Lectures on Sphere Arrangements - the Discrete Geometric Side

Fields Institute Monographs 32 The Fields Institute for Research in Mathematical Sciences Károly Bezdek Lectures on Sp...

Author: Károly Bezdek | 2013

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Fields Institute Monographs 32 The Fields Institute for Research in Mathematical Sciences

Károly Bezdek

Lectures on Sphere Arrangements – the Discrete Geometric Side

Fields Institute Monographs VOLUME 32 The Fields Institute for Research in Mathematical Sciences Fields Institute Editorial Board: Carl R. Riehm, Managing Editor Edward Bierstone, Director of the Institute Matheus Grasselli, Deputy Director of the Institute James G. Arthur, University of Toronto Kenneth R. Davidson, University of Waterloo Lisa Jeffrey, University of Toronto Barbara Lee Keyfitz, Ohio State University Thomas S. Salisbury, York University Noriko Yui, Queen’s University

The Fields Institute is a centre for research in the mathematical sciences, located in Toronto, Canada. The Institutes mission is to advance global mathematical activity in the areas of research, education and innovation. The Fields Institute is supported by the Ontario Ministry of Training, Colleges and Universities, the Natural Sciences and Engineering Research Council of Canada, and seven Principal Sponsoring Universities in Ontario (Carleton, McMaster, Ottawa, Toronto, Waterloo, Western and York), as well as by a growing list of Affiliate Universities in Canada, the U.S. and Europe, and several commercial and industrial partners. For further volumes: http://www.springer.com/series/10502

K´aroly Bezdek

Lectures on Sphere Arrangements—the Discrete Geometric Side

The Fields Institute for Research in the Mathematical Sciences

123

K´aroly Bezdek Department of Mathematics and Statistics University of Calgary Calgary, AB, Canada

ISSN 1069-5273 ISSN 2194-3079 (electronic) ISBN 978-1-4614-8117-1 ISBN 978-1-4614-8118-8 (eBook) DOI 10.1007/978-1-4614-8118-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013942302 Mathematics Subject Classification (2010): 52-02, 52C07, 52C10, 52C17, 52C22, 52C25, 52C35, 52C45, 52C99, 52B10, 52B11, 52B60, 52B99, 52A05, 52A21, 52A38, 52A40, 52A55, 52A99 © Springer International Publishing Switzerland 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Cover illustration: Drawing of J.C. Fields by Keith Yeomans Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

My parents encouraged all my interests. This book is dedicated to them with admiration.

Preface

The thematic program on “Discrete Geometry and Applications” took place at the Fields Institute for Research in Mathematical Sciences in Toronto between July 1 and December 31, 2011. The core part of the book is based on my three lectures, delivered at the the Fields Institute under the titles “Contact numbers for congruent sphere packings” (September 23, 2011), “Rigid ball-polyhedra” (October 11, 2011), and “On a strong version of the Kepler conjecture” (November 17, 2011). One can briefly describe discrete geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. J. Kepler was the first to raise discrete geometry problems on packings of balls in the early 1610s, but the systematic research began in the late 1940s with the work of L. Fejes T´oth. The Hungarian school he founded focused mainly on packing and covering problems, while a number of great mathematicians helped to lay a broad foundation for the emerging field of discrete geometry, including H. S. M. Coxeter, J. H. Conway, B. N. Delaunay, B. Gr¨unbaum, V. Klee, and C. A. Rogers. Two active areas that have been outstanding from the birth of discrete geometry are dense sphere packings and tilings. Both occupy a substantial part of my book as well. There is a chapter on unit sphere packings and there are a number of sections that apply the method of Delaunay and Voronoi tilings in the solutions of a variety of problems. Sphere packings and tilings have a very strong connection to number theory, coding, groups, and mathematical programming. In particular, the latter greatly helped to achieve recent breakthrough results. Extending the tradition of studying packings of spheres, another core topic of my book is the investigation of the monotonicity of volume under contractions of arbitrary arrangements of spheres. The research on this fundamental topic started with the conjecture of E. T. Poulsen and M. Kneser in the late 1950s. The third major topic of my book can be found under the sections on ball-polyhedra introducing an extension of the theory of convex polyhedral sets to the family of intersections of congruent balls. This part of my book is connected in many ways to the above mentioned major topics and it is also connected to some other important research areas as well including the one on coverings by planks (with close ties to geometric analysis). This topic is the forth major one in my book

vii

viii

Preface

discussed under coverings by cylinders. The research work on the latter topic started with a conjecture of A. Tarski in the early 1930s1 . My book is aimed at advanced undergraduate and early graduate students, as well as interested researchers. In addition to leading the reader to the frontiers of geometric research on sphere arrangements, it gives a short introduction to the relevant modern parts of discrete geometry. I have structured the book in such a way that the four major research topics (unit sphere packings, contractions of sphere arrangements, ball-polyhedra, and coverings by cylinders) are surveyed in individual chapters (Chaps. 1, 3, 5, and 7) each followed by a chapter with a collection of selected proofs (Chaps. 2, 4, 6, and 8). The survey chapters are readable independently from each other. The selected proofs combine elementary and convex geometry with analytic and in some cases, probabilistic or topological ideas. They are the results of the author’s joint work with a number of discrete geometers. In addition, an independently understandable collection of unsolved problems is compiled in Chap. 9. I am very much indebted to all my students and colleagues who attended my lectures and actively participated in the discussions at the Fields Institute in the fall of 2011. Furthermore, I want to thank the support of a number of colleagues and friends in particular, Ted Bisztriczky (Univ. of Calgary, Canada), K´aroly B¨or¨oczky (E¨otv¨os Univ., Hungary), Robert Connelly (Cornell Univ., USA), Bal´azs Csik´os (E¨otv¨os Univ., Hungary), Antoine Deza (McMaster Univ., Canada), G´abor Fejes T´oth (R´enyi Inst., Hungary), Herbert Edelsbrunner (Duke Univ., USA and IST, Austria), Ferenc Friedler (Univ. of Pannonia, Hungary), Thomas C. Hales (Univ. of Pittsburgh, USA), J´anos Pach (EPFL, Switzerland and R´enyi Inst., Hungary), Konrad Swanepoel (LSE, UK), Salvatore Torquato (Princeton Univ., USA), Asia I. Weiss (York Univ., Canada), and Yinyu Ye (Stanford Univ., USA). It is a particular pleasure for me to acknowledge my long-lasting research collaboration with my brother Andr´as Bezdek (Auburn Univ., USA and R´enyi Inst., Hungary) as well as with my friends Ted Bisztriczky (Univ. of Calgary, Canada) and Bob Connelly (Cornell Univ., USA). Also, it is a pleasure to acknowledge the excellent support provided by the Fields Institute; in particular, I would like to offer special thanks to Edward Bierstone, Alison Conway, Claire Dunlop, Matheus Grasselli, Debbie Iscoe, Matthias Neufang, and Carl Riehm. Also, special thanks are due to Samuel Reid (Univ. of Calgary, Canada) for the expressive drawings. Last but not least, I wish to thank my three sons, D´aniel, M´at´e, and M´ark, and in particular, my wife, ´ Eva, whose strong support and encouragement helped me a great deal during the long hours of writing. Calgary, AB, Canada

1

K´aroly Bezdek, Canada Research Chair

The author was supported by the Canada Research Chair program as well as a Natural Sciences and Engineering Research Council of Canada Discovery Grant.

Contents

1

Unit Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 The Contact Number Problem of Finite Sphere Packings . . . . . . . . . . . 1.2 Lower Bounds for Voronoi Cells in Sphere Packings .. . . . . . . . . . . . . . . 1.3 Dense Sphere Packings in Euclidean 3-Space.. . .. . . . . . . . . . . . . . . . . . . . 1.4 On Sphere Packings in High Dimensions . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Uniformly Stable and Periodic Extreme Lattices . . . . . . . . . . . . . . . . . . . .

1 1 5 9 12 14

2 Proofs on Unit Sphere Packings .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Proof of Theorem 1.1.6 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 An Upper Bound for Sphere Packings: Proof of (i) . . . . . . . . 2.1.2 An Upper Bound for the fcc Lattice: Proof of (ii) .. . . . . . . . . 2.1.3 Octahedral Unit Sphere Packings: Proof of (iii) . . . . . . . . . . . . 2.2 Proof of Theorem 1.2.4 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 The Voronoi Star of Voronoi Cells . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Estimating the Volume of a Voronoi Star from Below . . . . . 2.3 Proof of Theorem 1.2.5 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Metric Properties of Voronoi Cells . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Wedges of Types I, II, and III, and Truncated Wedges .. . . . 2.3.3 The Lemma of Comparison . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Volume Formulas for (Truncated) Wedges . . . . . . . . . . . . . . . . . 2.3.5 The Integral Representation of Surface Density .. . . . . . . . . . . 2.3.6 Truncation of Wedges Increases the Surface Density . . . . . . 2.3.7 Maximum Surface Density in Truncated Wedges of Type I. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.8 Surface Density in Truncated Wedges of Type II.. . . . . . . . . . 2.3.9 The Overall Estimate of Surface Density in Voronoi Cells . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Proof of Theorem 1.3.3 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Average Surface Area of Cells in Normal Tilings of a Cube .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 Average Surface Area of Cells in Normal Tilings . . . . . . . . . .

17 17 18 23 25 26 26 27 28 28 29 32 34 34 37 38 41 42 43 43 45 ix

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2.5

Proof of Theorem 1.3.5 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.1 Average Edge Curvature of Cells in Normal Tilings . . . . . . . 2.5.2 Average Edge Curvature of Voronoi Cells in Unit Ball Packings . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Proof of Theorem 1.5.3 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 The Signed Volume of Convex Polytopes . . . . . . . . . . . . . . . . . . 2.6.2 The Volume Force of Convex Polytopes .. . . . . . . . . . . . . . . . . . . 2.6.3 Critical Volume Condition .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.4 Strictly Locally Volume Expanding Convex Polytopes . . . . 2.6.5 From Critical Volume Condition to Uniform Stability .. . . .

49 50 50 51 52 54 56

3 Contractions of Sphere Arrangements . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 The Kneser–Poulsen Conjecture .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Kneser–Poulsen Conjecture for Continuous Contractions .. . . . . 3.3 On the Kneser–Poulsen Conjecture for General Contractions.. . . . . . 3.4 Kneser–Poulsen-Type Theorems in Non-Euclidean Spaces . . . . . . . . . 3.5 The Kneser–Poulsen Conjecture for Large Equal Radii . . . . . . . . . . . . . 3.6 Alexander’s Conjecture Revisited . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

57 57 58 60 62 64 65

4 Proofs on Contractions of Sphere Arrangements . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Proof of Theorem 3.3.2: Weighted Surface Volume . . . . . . . . . . . . . . . . . 4.2 Proof of Theorem 3.3.3: Codimension Two Volume .. . . . . . . . . . . . . . . . 4.3 Proof of Theorem 3.3.4: The Leapfrog Lemma ... . . . . . . . . . . . . . . . . . . . 4.4 Proof of Theorem 3.4.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 The Spherical Leapfrog Lemma.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Smooth Contractions via Schl¨afli’s Differential Formula .. 4.4.3 From Higher- to Lower-Dimensional Spherical Volume .. . 4.4.4 Putting Pieces Together .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Proof of Theorem 3.4.5 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.1 Monotonicity of the Volume of Hyperbolic Simplices .. . . . 4.5.2 Smooth One-Parameter Family of Hyperbolic Polyhedra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

67 67 71 73 74 74 75 75 76 77 77

5 Ball-Polyhedra and Spindle Convex Bodies . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Intersections of Congruent Balls . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Disk-Polygons Revisited . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 On a Steinitz-Type Problem for Ball-Polyhedra .. . . . . . . . . . . . . . . . . . . . 5.4 On Global and Local Rigidity of Ball-Polyhedra . . . . . . . . . . . . . . . . . . . . 5.5 Separation and Support for Spindle Convex Bodies . . . . . . . . . . . . . . . . . 5.6 A Carath´eodory-Type Theorem for Spindle Convex Hulls . . . . . . . . . . 5.7 Illuminating Ball-Polyhedra and Spindle Convex Bodies . . . . . . . . . . . 5.8 An Euler–Poincar´e-Type Formula for Ball-Polyhedra .. . . . . . . . . . . . . .

83 83 84 88 89 93 94 94 97

2.6

48 48

81

6 Proofs on Ball-Polyhedra and Spindle Convex Bodies . . . . . . . . . . . . . . . . . . . 99 6.1 Proof of Theorem 5.2.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 6.1.1 On Translates of the Interior of a Convex Body .. . . . . . . . . . . 99 6.1.2 From Generalized Billiard Trajectories to Shortest Ones . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101

Contents

6.2 6.3 6.4 6.5 6.6

xi

Proof of Theorem 5.4.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Proof of Theorem 5.4.2 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Underlying Truncated Delaunay Complex of a Ball-Polyhedron . . . Proof of Theorem 5.4.6 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Infinitesimally Rigid Polyhedron and Dual Ball-Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 Proof of Theorem 5.4.10 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.8 Proofs of Theorems 5.5.1–5.5.3.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.8.1 Strict Separation by Spheres of Radii at Most One.. . . . . . . . 6.8.2 Characterizing Spindle Convex Sets . . . .. . . . . . . . . . . . . . . . . . . . 6.8.3 Separating Spindle Convex Sets . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.9 Proof of Theorem 5.6.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.9.1 Spindle Convex Hulls and Supporting Spheres.. . . . . . . . . . . . 6.9.2 Carath´eodory’s Theorem for Spindle Convex Hulls. . . . . . . . 6.10 Proof of Theorem 5.7.3 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.10.1 On the Boundary of Spindle Convex Hulls . . . . . . . . . . . . . . . . . 6.10.2 On the Euclidean Diameter of Spindle Convex Hulls .. . . . . 6.10.3 An Upper Bound Based on a Probabilistic Approach .. . . . . 6.10.4 Schramm’s Lower Bound .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.10.5 On Sets of Given Diameter to Cover Spherical Space . . . . . 6.10.6 The Final Upper Bound for the Illumination Number.. . . . . 6.11 Proof of Theorem 5.7.5 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.12 Proof of Theorem 5.8.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.12.1 The CW-Decomposition of a Standard Ball-Polyhedron . . 6.12.2 On the Number of Generating Balls . . . .. . . . . . . . . . . . . . . . . . . . 6.12.3 On the Face Lattices of Standard Ball-Polyhedra . . . . . . . . . .

103 104 106 109 110 111 114 114 115 115 116 116 117 118 118 120 121 122 124 125 126 126 127 127 128

7 Coverings by Cylinders.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Plank Theorems: Old and New . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Covering Convex Bodies by Cylinders . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Kadets–Ohmann-Type Theorems .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 On Partial Coverings of Balls by Planks. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

131 131 136 137 139

8 Proofs on Coverings by Cylinders. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Proof of Theorem 7.1.8 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1.1 On Coverings of Convex Bodies by Two Planks . . . . . . . . . . . 8.1.2 Minimizing the Greatest mth Successive C-Inradius . . . . . . 8.2 Proof of Theorem 7.1.12 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2.1 On an Extension of a Helly-Type Result of Klee .. . . . . . . . . . 8.2.2 On Some Concave Functions of Successive Inradii .. . . . . . . 8.2.3 Estimating Sums of Successive Inradii .. . . . . . . . . . . . . . . . . . . . 8.3 Proof of Theorem 7.2.2 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3.1 Covering Ellipsoids by 1-Codimensional Cylinders . . . . . . . 8.3.2 Covering Convex Bodies by Cylinders .. . . . . . . . . . . . . . . . . . . . 8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes .. . . . . . . . . . . . . . .

143 143 143 144 145 146 147 148 149 149 150 151

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8.5 8.6

Proof of Theorem 7.4.2: Partial Coverings in Euclidean 3-Space .. . 155 Proof of Theorem 7.4.6: Lower Bounds for Ball-Polyhedra.. . . . . . . . 155

9 Research Problems: An Overview . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Unit Sphere Packings .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.1 The Contact Number Problem of Finite Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1.2 Dense Sphere Packings in Euclidean 3-Space .. . . . . . . . . . . . . 9.1.3 On Sphere Packings in High Dimensions .. . . . . . . . . . . . . . . . . . 9.1.4 Uniformly Stable and Periodic Extreme Lattices.. . . . . . . . . . 9.2 Contractions of Sphere Arrangements .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.1 The Kneser–Poulsen Conjecture . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.2 Kneser–Poulsen-Type Theorems in Non-Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2.3 Alexander’s Conjecture Revisited . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Ball-Polyhedra and Spindle Convex Bodies . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.1 Disk-Polygons Revisited . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3.2 On a Steinitz-Type Problem for Ball-Polyhedra .. . . . . . . . . . . 9.3.3 On Global and Local Rigidity of Ball-Polyhedra.. . . . . . . . . . 9.3.4 Illuminating Ball-Polyhedra and Spindle Convex Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Coverings by Cylinders.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.1 Plank Theorems: Old and New . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.2 Covering Convex Bodies by Cylinders .. . . . . . . . . . . . . . . . . . . . 9.4.3 Kadets–Ohmann-Type Theorems . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4.4 On Partial Coverings of Balls by Planks .. . . . . . . . . . . . . . . . . . .

157 157 157 157 158 158 159 159 159 160 160 160 161 162 162 163 163 163 164 164

Glossary . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 165 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 169

Chapter 1

Unit Sphere Packings

Abstract Unit sphere packings are the classical core of (discrete) geometry. We survey old as well new results giving an overview of the art of the matters. The emphases are on the following five topics: the contact number problem (generalizing the problem of kissing numbers), lower bounds for Voronoi cells (studying Voronoi cells from volumetric point of view), dense sphere packings in Euclidean 3-space (studying a strong version of the Kepler conjecture), sphere packings in Euclidean dimensions higher than 3, and uniformly stable sphere packings.

1.1 The Contact Number Problem of Finite Sphere Packings Let B be a d -dimensional ball in the d -dimensional Euclidean space Ed . As is well known, a finite packing of B in Ed is a finite family of non-overlapping congruent copies of B in Ed . Furthermore, the contact graph of a finite packing of B in Ed is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if and only if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of n non-overlapping translates of the given Euclidean ball B can have in Ed . Harborth [124] proved the following remarkable result on the contact graphs of congruent circular disk packings in E2 . Theorem 1.1.1. The maximum number of touching pairs in a packing of n congruent circular disks in E2 is precisely b3n

p 12n 3c:

The analogue question in the hyperbolic plane has been studied by Bowen in [66]. We prefer to quote his result in the following geometric way. K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 1, © Springer International Publishing Switzerland 2013

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1 Unit Sphere Packings

Theorem 1.1.2. Consider circle packings in the hyperbolic plane, by finitely many congruent circles, which maximize the number of touching pairs for the given number of congruent circles. Then such a packing must have all of its centers located on the vertices of a triangulation of the hyperbolic plane by congruent equilateral triangles, provided the diameter D of the circles is such that an equilateral triangle in the hyperbolic plane of side length D has each of its angles equal to 2 N for some N > 6. It is not hard to see that one can extend the above result to the 2-dimensional spherical space S2 exactly in the way as the above phrasing suggests. However, we get a more general approach if we do the following. Take n non-overlapping unit diameter balls in a convex position in E3 ; that is, assume there exists a 3-dimensional convex polyhedron whose vertices are center points with each additional center point sitting on the boundary of that convex polyhedron, where n 4 is a given integer. Obviously, the shortest distance among the center points is at least one. Then count the unit distances showing up between pairs of center points but count only those pairs that generate a unit line segment on the boundary of the given 3-dimensional convex polyhedron. Finally, maximize this number for the given n and label this maximum by c.n/. In the following statement of D. Bezdek [34] the convex polyhedra entering are called generalized deltahedra (or in short, g-deltahedra) (see also [37]) mainly because that family of convex polyhedra includes all deltahedra classified quite some time ago by Freudenthal and van der Waerden in [105]. Theorem 1.1.3. c.n/ 3n 6, where equality is attained for infinitely many n namely, for those for which there exists a 3-dimensional convex polyhedron each face of which is an edge-to-edge union of some regular triangles of side length 1 such that the total number of generating regular triangles on the boundary of the convex polyhedron is precisely 2n 4 with a total number of 3n 6 sides of length 1 and with a total number of n vertices. Theorem 1.1.3 proposes to find a proper classification for g-deltahedra, a question that is still open. Some partial results on that can be found in the recent paper [37] of D. Bezdek, the main result of which states that the regular icosahedron has the smallest isoperimetric quotient among all g-deltahedra. For the sake of completeness we mention that this result supports the still open icosahedral conjecture of Steiner (1841), according to which among all convex polyhedra isomorphic to an icosahedron (i.e. having the same face structure as an icosahedron) the regular icosahedron has the smallest isoperimetric quotient. Another interesting result on g-deltahedra was obtained by D. Bezdek in [34]. It claims that every g-deltahedron has an edge unfolding. This is part of the general problem, raised by Shephard (1975) and motivated also by some drawings of D¨urer (1525), of whether every convex polyhedron has an edge unfolding, that is can be cut along some of its edges and then folded into a single planar polygon without overlap. For more details on this we refer the interested reader to the lavishly illustrated book [93] of Demaine and O’Rourke.

1.1 The Contact Number Problem of Finite Sphere Packings

3

Now, we are ready to phrase the contact number problem of finite congruent sphere packings in E3 . For a given positive integer n 2 find the largest number C.n/ of touching pairs in a packing of n congruent balls in E3 . One can regard this problem as a combinatorial relative of the Kepler conjecture on the densest unit sphere packings in E3 . It is natural to continue with the following question. Problem 1.1.4. Find those positive integers n for which C.n/ can be achieved in a packing of n unit balls in E3 consisting of parallel layers of unit balls each being a subset of the densest infinite hexagonal layer of unit balls. Harborth’s result [124] (see Theorem 1.1.1) implies in a straightforward way that if the maximum number of touching pairs in packings of n congruent circular disks in E2 is denoted by c .n/, then p 3n c .n/ p D 12 D 3:464 : : : : n!C1 n lim

The author [33] has proved the following estimates in higher dimensions. The number of touching pairs in an arbitrary packing of n > 1 unit balls in Ed , d 3 is less than 1 d 1 d 1 1 d n d ı d d n d ; 2 2 where d stands for the kissing number of a unit ball in Ed (i.e., it denotes the maximum number of non-overlapping unit balls of Ed that can touch a given unit ball in Ed ) and ıd denotes the largest possible density for (infinite) packings of unit balls in Ed . Now, recall that on the one hand, according to the well-known theorem of Kabatiansky and Levenshtein [133] d 20:401d.1Co.1// and ıd 20:599d.1Co.1// as d ! C1 on the other hand, 3 D 12 (for the first complete proof see [170]) moreover, according to the recent breakthrough result of Hales [114] ı3 D p . 18 Thus, by combining the above results together we get that the number of touching pairs in an arbitrary packing of n > 1 unit balls in Ed is less than d 1 1 1 0:401d.1Co.1// 2 n 20:401.d 1/.1o.1// n d 2 2

as d ! C1 and in particular, it is less than 1 6n 8

p 18

23

2

2

n 3 D 6n 0:152 : : : n 3

for d D 3. The main purpose of this section is to report on recent improvements of the latter estimate. In order, to state those results in a proper form we need to introduce a bit of additional terminology. If P is a packing of n unit balls in E3 , then let C.P/ stand for the number of touching pairs in P, that is, let C.P/ denote the number of edges of the contact graph of P and call it the contact number of P. Moreover, let C.n/ be the largest C.P/ for packings P of n unit balls in E3 . Finally,

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1 Unit Sphere Packings

let us imagine that we generate packings of n unit balls in E3 in such a special way that each and every center of the n unit balls chosen, is a lattice point of the face-centered cubic lattice f cc with shortest non-zero lattice vector of length 2. Then let Cf cc .n/ denote the largest possible contact number of all packings of n unit balls obtained in this way. First, recall that according to [114] the lattice unit sphere packing generated by f cc gives the largest possible density for unit ball packings in E3 , namely p with each ball touched by 12 others such that their 18 centers form the vertices of a cuboctahedron. Second, it is easy to see that Cf cc .2/ D C.2/ D 1; Cf cc .3/ D C.3/ D 3; Cf cc .4/ D C.4/ D 6. Third, it is natural to conjecture that Cf cc .9/ D C.9/ D 21. Based on the trivial inequalities C.n C 1/ C.n/ C 3; Cf cc .n C 1/ Cf cc .n/ C 3 valid for all n 2, it would follow that Cf cc .5/ D C.5/ D 9; Cf cc .6/ D C.6/ D 12; Cf cc .7/ D C.7/ D 15, and Cf cc .8/ D C.8/ D 18. In general, clearly C.n/ 3n 6. Furthermore, we note that C.10/ 25; C.11/ 29, and C.12/ 33. In order, to see that one should take the union U of two regular octahedra of edge length 2 in E3 such that they share a regular triangle face T in common and lie on opposite sides of it. If we take the unit balls centered at the 9 vertices of U, then there are exactly 21 touching pairs among them. Also, we note that along each side of T the dihedral angle of U is concave and in fact, it can be completed to 2 by adding twice the dihedral angle of a regular tetrahedron in E3 . This means that along each side of T two triangular faces of U meet such that for their four vertices there exists precisely one point in E3 lying outside U and at distance 2 from each of the four vertices. Finally, if we take the 12 vertices of a cuboctahedron of edge length 2 in E3 along with its center of symmetry, then the 13 unit balls centered about them have 36 contacts implying that C.13/ 36. Whether in any of the inequalities C.10/ 25; C.11/ 29; C.12/ 33, and C.13/ 36 we have equality is a challanging question. In connection with this problem we call the reader’s attention to the very recent article of Hayes [126]. It gives an overview of the computational methods presented in the papers [10] and [129] that are based on exhaustive enumeration and elementary geometry. According to [126] the main results stated in those papers are: C.9/ D 21 ([10]), C.10/ D 25 ([10]) and C.11/ D 29 ([129]). However, the status of those claims remains to be seen. Last but not least, as a natural extension of the 3-dimensional kissing number problem one is tempted to raise the following particular question (see also the relevant discussion in [10]). Conjecture 1.1.5. C.12/ D 33 and C.13/ D 36. For C.n/ in general, when n is an arbitrary positive integer, we have the following estimates proved in [42] and [54]. Theorem 1.1.6. 2

3 for all n 2. (i) C.n/ < 6n 0:926n p 2 3 3 18 23 (ii) Cf cc .n/ < 6n n D 6n 3:665 : : : n 3 for all n 2. p 2 2 (iii) 6n 3 486n 3 < Cf cc .n/ C.n/ for all n D k.2k3 C1/ with k 2.

As an immediate result we get

1.2 Lower Bounds for Voronoi Cells in Sphere Packings

5

Corollary 1.1.7. 0:926 < for all n D

k.2k 2 C1/ 3

6n C.n/ n

2 3

<

p 3 486 D 7:862 : : :

with k 2.

The latter claim leads us to a rather basic question. Problem 1.1.8. Does the limit limn!C1

6nC.n/ 2

exist?

n3

The following was noted in [33]. Due to the Minkowski difference body method (see for example, Chap. 6 in [160]) the family PK WD ft1 CK; t2 CK; : : : ; tn CKg of n translates of the convex body K in Ed is a packing if and only if the family PKo WD ft1 C Ko ; t2 C Ko ; : : : ; tn C Ko g of n translates of the symmetric difference body Ko WD 12 .K C .K// of K is a packing in Ed . Moreover, the number of touching pairs in the packing PK is equal to the number of touching pairs in the packing PKo . Thus, for this reason and for the reason that if K is a convex body of constant width in Ed , then Ko is a ball of Ed , Theorem 1.1.6 extends in a straightforward way to translative packings of convex bodies of constant width in E3 . For the sake of completeness we mention that the nature of contact numbers changes dramatically for non-congruent sphere packings in E3 . For more details on that we refer the interested reader to the elegant paper [141] of Kuperberg and Schramm.

1.2 Lower Bounds for Voronoi Cells in Sphere Packings A family of non-overlapping 3-dimensional balls of radii 1 in Euclidean 3-space, E3 is called a unit ball packing in E3 : The density of the packing is the proportion of space covered by these unit balls. The sphere packing problem asks for the densest packing of unit balls in E3 . The conjecture that the density of any unit ball packing in E3 is at most p18 D 0:74078 : : : is often attributed to Kepler’s statement of 1611. The problem of proving the Kepler conjecture appears as part of Hilbert’s 18th problem [127]. Using an ingenious argument which works in any dimension, Rogers [159] obtained the upper bound 0:77963 : : : for the density of unit ball packings in E3 : This bound has been improved by Lindsey [144], and Muder [150, 151] to 0:773055 : : : : Hsiang [130, 131] proposed an elaborate line of attack (along the ideas of L. Fejes T´oth suggested 40 years earlier), but his claim that he settled Kepler’s conjecture seems exaggerated. However, so far no one has found any gap in the approach of Hales [103, 112–118, 121] although no one has been able to fully verify it either. This is not too surprising, given that the detailed argument is described in several papers and relies on long computer-aided calculations of more than 5,000 subproblems. Thus, after several years of extremely hard work, Hales with the help of Ferguson [103] has been able to finish his complex

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project and they were able to come up with the most detailed and complete-looking computer-supported proof of the longstanding conjecture of Kepler that is presently known. We summarize their heroic achievement in a short statement. Theorem 1.2.1. The densest packing of unit balls in E3 has density attained by the “cannonball packing”.

p , 18

which is

For several of the above-mentioned papers Voronoi cells of unit ball packings play a central role. Recall that the Voronoi cell of a unit ball in a packing of unit balls in E3 is the set of points that are not farther away from the center of the given ball than from any other ball’s center. As is well known, the Voronoi cells of a unit ball packing in E3 form a tiling of E3 . One of the most attractive problems on Voronoi cells is the dodecahedral conjecture of L. Fejes T´oth published in [98]. According to this the volume of any Voronoi cell in a packing of unit balls in E3 is at least as large as the volume of a regular dodecahedron with inradius 1. Very recently Hales and McLaughlin [122,123] have succeeded in proving this long-standing conjecture of L. Fejes T´oth. Thus, we have the following theorem. Theorem 1.2.2. The volume of any Voronoi cell in a packing of unit balls in E3 is at least as large as the volume of a regular dodecahedron with inradius 1. We wish to mention that although neither Theorem 1.2.1 nor Theorem 1.2.2 implies the other, their proofs follow a similar outline and share a significant number of methods, and in particular, both are based on long computer calculations. Just like in E3 , the Voronoi cell of a unit ball in a packing of unit balls in Ed is the set of points that are not farther away from the center of the given ball than from any other ball’s center. As is well known, the Voronoi cells of a unit ball packing in Ed form a tiling of Ed . One of the most attractive results on the sphere packing problem was proved by C. A. Rogers [159] in 1958. It was rediscovered by Baranovskii [16] and extended to spherical and hyperbolic spaces by B¨or¨oczky [67]. It can be phrased as follows. Take a regular d -dimensional simplex of edge length 2 in Ed and then draw a d -dimensional unit ball around each vertex of the simplex. Let d denote the ratio of the volume of the portion of the simplex covered by balls to the volume of the simplex. Theorem 1.2.3. The volume of any Voronoi cell in a packing of unit balls in Ed is at least !dd , where !d denotes the volume of a d -dimensional unit ball. The following strengthening of Theorem 1.2.3 has been proved by the author in [29]. (See also [57] for a somewhat simpler proof.) Theorem 1.2.4. The surface volume of any Voronoi cell in a packing of unit balls d in Ed ; d 2 is at least d! . d Indeed, Theorem 1.2.4 implies Theorem 1.2.3 by observing that the volume of a Voronoi cell in a packing of unit balls in Ed is at least as large as d1 times the surface volume of the Voronoi cell in question. The next theorem due to the author [32] improves the estimate of Theorem 1.2.4 even further for all d 8. For this we

1.2 Lower Bounds for Voronoi Cells in Sphere Packings

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need a bit of notation. As usual, let lin./; aff./; conv./; vold ./; !d , Svold 1 ./, dist.; /, k k, and o refer to the linear hull, the affine hull, the convex hull in Ed , the d -dimensional Euclidean volume measure, the d -dimensional volume of a d -dimensional unit ball, the .d 1/-dimensional spherical volume measure, the distance function in Ed , the standard Euclidean norm, and to the origin in Ed . Let convfo; w1 ; : : : ; wd g be a d -dimensional simplex having the property that the linear hull linfwj wi ji < j d g is orthogonal to the vector wi in Ed , d 8 for all 1 i d 1; that is, let convfo; w1 ; : : : ; wd g be a d -dimensional orthoscheme in Ed . Moreover, let r 2i for all 1 i d: kwi k D i C1 It is clear that in the right triangle M wd 2 wd 1qwd with right q angle at the vertex 2 2 D kwd 1 wd 1 we have the inequality kwd wd 1 k D d.d C1/ < .d 1/d wd 2 k and therefore †wd 1 wd 2 wd < of the triangle M wd 2 wd 1 wd let

4.

Now, in the plane afffwd 2 ; wd 1 ; wd g

C wd 2 wd wd C1 denote the circular sector of central angle †wd wd 2 wd C1 D

†wd 1 wd 2 wd 4

and of center wd 2 sitting over the circular arc with endpoints wd ; wd C1 and radius kwd wd 2 k D kwd C1 wd 2 k such that C wd 2 wd wd C1 and M wd 2 wd 1 wd are adjacent along the line segment wd 2 wd and are separated by the line of wd 2 wd . Then let D.wd 2 ; wd 1 ; wd ; wd C1 / DM wd 2 wd 1 wd [ C wd 2 wd wd C1 be the convex domain generated by the triangle M wd 2 wd 1 wd with constant angle †wd 1 wd 2 wd C1 D

: 4

Now, let W D conv fo; w1 ; : : : ; wd 3 g [ D.wd 2 ; wd 1 ; wd ; wd C1 / be the d -dimensional wedge (or cone) with .d 1/-dimensional base QW D conv fw1 ; : : : ; wd 3 g [ D.wd 2 ; wd 1 ; wd ; wd C1 / and apex o:

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Fig. 1.1 The density bound O d

Ed wd wd+1

wd–1

W wd–2 o

Finally, if B D fx 2 Ed j dist.o; x/ D kxk 1g denotes the d -dimensional unit ball centered at the origin o of Ed and Sd 1 D fx 2 Ed j dist.o; x/ D kxk D 1g denotes the .d 1/-dimensional unit sphere centered at o, then let (Fig. 1.1) O d D

vold .W \ B/ Svold 1 .W \ Sd 1 / D vold 1 .QW / vold .W/

be the surface density (resp., volume density) of the unit sphere Sd 1 (resp., of the unit ball B) in the wedge W . For the sake of completeness we remark that as the regular d -dimensional simplex of edge length 2 can be dissected into .d C1/Š pieces each being congruent to convfo; w1 ; : : : ; wd g therefore d D

vold .convfo; w1 ; : : : ; wd g \ B/ : vold .convfo; w1 ; : : : ; wd g/

Now, we are ready to state the main result of [32]. Recall that the surface density of any unit sphere in its Voronoi cell in a unit sphere packing of Ed is defined as the ratio of the surface volume of the unit sphere to the surface volume of its Voronoi cell. Theorem 1.2.5. The surface volume of any Voronoi cell in a packing of unit balls d in the d -dimensional Euclidean space Ed , d 8 is at least d! ; that is, the surface O d density of any unit sphere in its Voronoi cell in a unit sphere packing of Ed , d 8 is at most O d . Thus, the volume of any Voronoi cell in a packing of unit balls in Ed , d 8 is at least !O dd and so, the (upper) density of any unit ball packing in Ed , d 8 is at most O d (< d ). Last but not least we note that the proof of Theorem 1.2.5 published in [32] gives a proof of the following even stronger statement. Take a Voronoi cell of a unit ball in a packing of unit balls in the d -dimensional Euclidean space Ed ; d 8 and then take theqintersection of the given Voronoi cell with the closed d -dimensional ball of radius

2d d C1

concentric to the unit ball of the Voronoi cell. Then the surface area

of the truncated Voronoi cell is at least

d !d O d

.

1.3 Dense Sphere Packings in Euclidean 3-Space

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1.3 Dense Sphere Packings in Euclidean 3-Space The central problem that we raise in this section can be phrased informally as follows. Problem 1.3.1. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In order to state our problem in more precise terms we proceed as follows. Let T be a tiling of the 3-dimensional Euclidean space E3 into convex polyhedra Pi ; i D 1; 2; : : : each containing a unit ball say, Pi containing the closed 3-dimensional ball Bi having radius 1 for i D 1; 2; : : : . Also, we assume that there is a finite upper bound for the diameters of the convex cells in T , i.e., supfdiam.Pi /ji D 1; 2; : : : g < 1, where diam./ denotes the diameter of the corresponding set. In short, we say that T is a normal tiling of E3 with the underlying packing P of the unit balls Bi ; i D 1; 2; : : : . Then we define the (lower) average surface area s.T / of the cells in T as follows: P fi jBi CL g sarea.Pi \ CL / s.T / WD lim inf ; L!1 cardfi jBi CL g where CL denotes the cube centered at the origin o with edges parallel to the coordinate axes of E3 and having edge length L furthermore, sarea./ and card./ denote the surface area and cardinality of the corresponding sets. (We note that it is rather straightforward to show that s.T / is independent from the choice of the coordinate system of E3 .) There is very natural way to generate a large family of normal tilings. Namely, let PR be an arbitrary packing of unit balls in E3 with the property that each closed ball of radius R in E3 contains the center of at least one unit ball in PR . Recall that the Voronoi cell of a unit ball in PR is the set of points that are not farther away from the center of the given ball than from any other ball’s center. It is well known that the Voronoi cells in question form a tiling of E3 (for more details see [160]). Furthermore, the Voronoi tiling obtained in this way, is going to be a normal one because each Voronoi cell is contained in the closed ball of radius R concentric to the unit ball of the given Voronoi cell and therefore the diameter of each Voronoi cell is at most 2R. Also, we recall here the strong dodecahedral conjecture of [29]: the surface area of every (bounded) Voronoi cell in a packing of unit balls is at least that of a regular dodecahedron of inradius 1, i.e., it is at least 16:6508 : : : . After a sequence of partial results obtained in [29, 50], and [7] (proving the lower bounds 16:1433 : : : , 16:1445 : : : , and 16:1977 : : : ), just very recently, Hales [119] has announced a computer assisted proof of the strong dodecahedral conjecture. Thus, we have the following theorem. Theorem 1.3.2. The surface area of any Voronoi cell in a packing of unit balls in E3 is at least that of a regular dodecahedron of inradius 1, i.e., it is at least 16:6508 : : : .

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By adjusting Kert´esz’s volume estimation technique [136] to the problem on estimating surface area and making the necessary modifications, the author [43] proved the following inequality. Theorem 1.3.3. Let T be an arbitrary normal tiling of E3 . Then the average 24 surface area of the cells in T is always at least p , i.e., 3 24 s.T / p D 13:8564 : : : : 3 Most likely the lower bound in Theorem 1.3.3 can be improved further however, any such improvement would require additional new ideas. In particular, recall that in the face-centered cubic lattice packing of unit balls in E3 , when each ball is touched by 12 others, the Voronoi cells of the unit p balls are regular rhombic dodecahedra of inradius 1 and of surface area 12 2 (for more details on the geometry involved see [99]). Thus, it is immediate to raise the following question: prove or disprove that if T is an arbitrary normal tiling of E3 , then p s.T / 12 2 D 16:9705 : : : : (1.1) Let us mention that an affirmative answer to (1.1) (resp., a partially affirmative answer to (1.1) when only Voronoi tilings of unit ball packings are considered) would imply the Kepler conjecture. As is well known, the Kepler conjecture has been proved by Hales in a sequence of celebrated papers ([114, 116, 117, 121], and [118]) concluding that the density of any unit ball packing in E3 is at most p . 18 p Now, if s.T / 12 2 were true for arbitrary normal tiling T of E3 with the unit p balls of the cells forming the packing P in E3 (resp., if the inequality s.T / 12 2 were true for Voronoi tilings T of unit ball packings P in E3 ), then based on the obvious inequalities X

vol.Pi \ CL / vol.CL / and

fi jBi CL g

1 sarea.Pi \ CL / vol.Pi \ CL /; 3

(where vol./ denotes the volume of the corresponding set) we would get that the (upper) density ı.P/ WD lim supL!1 the inequality

4 3 cardfi jBi CL g vol.CL /

CL g fi jBi CL g vol.Pi \ CL /

ı.P/ lim sup P L!1

lim sup P L!1

of the packing P must satisfy

4 3 cardfi jBi

4 4cardfi jBi CL g D p s.T / 18 fi jBi CL g sarea.Pi \ CL /

1.3 Dense Sphere Packings in Euclidean 3-Space

11

Thus, one could regard the affirmative version of (1.1), stated for Voronoi tilings of unit ball packings, as a strong version of the Kepler conjecture: Conjecture 1.3.4. If Tpdenotes the Voronoi tiling of an arbitrary unit ball packing in E3 , then s.T / 12 2 D 16:9705 : : :. As an additional observation we mention that an affirmative answer to (1.1) would imply also the rhombic dodecahedral conjecture of [30]. According to that conjecture the surface area of any 3-dimensional parallelohedron of inradius at least 1 (i.e., the surface area of any convex polyhedron containing a unit ball and having p a family of translates tiling E3 ) is at least as large as 12 2 D 16:9705 : : :. By taking a closer look of the proof of Theorem 1.3.3 in [43] one can derive the following stronger estimate. In order to state it in a proper form we need to introduce some additional terminology. Let P be a convex polyhedron in E3 and let P E.P/ denote the family of the edges of P. Then ecurv.P/ WD e2E.P/ L.e/ tan ˛2e is called the edge curvature of P, where L.e/ stands for the length of the edge e 2 E.P/ and ˛e denotes the angle between the outer normal vectors of the two faces of P meeting along the edge e. (For more insight on edge curvature we refer the interested reader to p. 287 in [99].) Now, if T is an arbitrary normal tiling of E3 with the underlying packing P of the unit balls Bi ; i D 1; 2; : : : , then we define the (lower) average edge curvature ec.T / of the cells in T as follows: P ec.T / WD lim inf

fi jBi CL g

ecurv.Pi \ CL /

cardfi jBi CL g

L!1

:

Based on this we can state the following stronger version of Theorem 1.3.3. Theorem 1.3.5. Let T be an arbitrary normal tiling of E3 . Then 24 s.T / ec.T / p D 13:8564 : : : : 3 Moreover, if T is a Voronoi tiling of a unit ball packing in E3 , then 2 2 ec.T / p p D 14:6176 : : : : 6 6 arcsin p1 6 3

Thus, we have the following stronger version of our original problem. Problem 1.3.6. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average edge curvature of the cells? Last but not least, it is very tempting to further relax the conditions in our original problem by replacing convex cells with cells that are measurable and have measurable boundaries and ask the following more general question: if the Euclidean 3-space is partitioned into cells each containing a unit ball, how should

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1 Unit Sphere Packings

the shapes of the cells be designed to minimize the average surface area of the cells? One can regard this question as a foam problem, in particular, as a relative of Kelvin’s foam problem (on partitioning E3 into unit volume cells with minimum average surface area) since foams are simply tilings of space that (under proper conditions) minimize surface area. Although foams are well studied (see the relevant sections of the highly elegant book [149] of Morgan), it is far not clear what would be a good candidate for the proper minimizer in the foam question just raised. As a last note we mention that K. Brakke, Nov., 2011, Personal communication, by properly modifying the Williams foam, has just obtained a partition of the Euclidean 3-space into congruent cells each containing a unit ball and having surface area p 16:95753 < 12 2 D 16:9705 : : :.

1.4 On Sphere Packings in High Dimensions Recall that a family of non-overlapping d -dimensional balls of radii 1 in the d dimensional Euclidean space Ed is called a unit ball packing of Ed . The density of the packing is the proportion of space covered by these unit balls. The sphere packing problem asks for the densest packing of unit balls in Ed . Indubitably, of all problems concerning packing it was the sphere packing problem which attracted the most attention in the past decade. It has its roots in geometry, number theory, and information theory and it is part of Hilbert’s 18th problem. The reader is referred to [81] (especially the third edition, which has about 800 references covering 1988– 1998) for further information, definitions, and references. In what follows we report on a few selected recent developments. The lower bound for the volume of Voronoi cells in congruent sphere packings due to C. A. Rogers [159] and Baranovskii [16] (mentioned earlier) combined with Daniel’s asymptotic formula [160] yields the following corollary. Theorem 1.4.1. The (upper) density of any unit ball packing in Ed is at most d D

d .0:5Co.1//d 2 e

.as d ! 1/:

Then 20 years later, in 1978 Kabatiansky and Levenshtein [133] improved this bound in the exponential order of magnitude as follows. They proved the following theorem. Theorem 1.4.2. The (upper) density of any unit ball packing in Ed is at most 2.0:599Co.1//d

.as d ! 1/:

For the sake of completeness we mention also that, based on Theorem 1.2.5 as well as [81] (see p. 20), the upper bound d of Rogers is larger than the author’s upper bound O d for all 8 d 42 (in fact, d > O d for all d 8) and then for all d > 42 the author’s upper bound O d is larger than the Kabatiansky–Levenshtein upper bound.

1.4 On Sphere Packings in High Dimensions

13

There has been some very important recent progress concerning the existence of economical packings. On the one hand, improving earlier results, Ball [13] proved the following statement through a very elegant completely new variational argument. (See also [109] for a similar result of W. Schmidt on centrally symmetric convex bodies.) Theorem 1.4.3. For each d , there is a lattice packing of unit balls in Ed with density at least

where .d / D

P1

1 kD1 k d

d 1 .d /; 2d 1 is the Riemann zeta function.

In addition, when d is divisible by 4 Vance [176] has managed to improve the 3.d 1/ estimate of Theorem 1.4.3 by replacing 2dd1 1 with e2d 1 . Furthermore, just very recently Venkatesh [177] announced that Ball’s estimate can be improved further 2 by any factor less than sinh 2 .e/ D 32;981:9010 : : : for every sufficiently large d . e3 Finally, the possibility of some additional (in fact, exponential) improvements has been put forward as a conjecture by Torquato and Stillinger in [175]. On the other hand, for some small values of d , there are explicit packings which give (considerably) higher densities than the bounds just mentioned. The reader is referred to [81] and [153] for a comprehensive view of results of this type. Improvements on the upper bound d of Rogers and on the upper bound O d of the author for the dimensions from 4 to 36 have been obtained recently by Cohn and Elkies [76]. They developed an analogue for sphere packing of the linear programming bounds for error-correcting codes, and used it to prove new upper bounds for the density of congruent sphere packings, which led to the best upper bounds in dimensions between 4 and 36. (Just very recently further improvements have been obtained in [89] for dimensions 4; : : : ; 9 except 8 using semidefinite programming.) The method of Cohn and Elkies together with the best-known sphere packings yields the following nearly optimal estimates in dimensions 8 and 24. Theorem 1.4.4. The density of the densest unit ball packing in E8 (resp., E24 ) is at least 0:2536 : : : (resp., 0:00192 : : : ) and is at most 0:2537 : : : : (resp., 0:00196 : : : ). In fact, just very recently Cohn and Kumar [77] were able to prove the impressive estimate that no packing of congruent balls in E24 can exceed the Leech lattice’s density by a factor 1 C 1:65 1030 . Last but not least Cohn and Elkies [76] conjecture that their approach can be used to solve the sphere packing problem in E8 (resp., E24 ). Conjecture 1.4.5. The E8 root lattice (resp., the Leech lattice) that produces the corresponding lower bound in the previous theorem in fact, represents the largest possible density for unit sphere packings in E8 (resp., E24 ). If linear programming bounds can indeed be used to prove optimality of these lattices, it would not come as a complete surprise because, for example, the kissing number problem in these dimensions was solved similarly.

14

1 Unit Sphere Packings

Finally, we mention the following breakthrough result of Cohn and Kumar [77] according to which the Leech lattice is the densest lattice packing in E24 . (The densest lattices have been known up to dimension 8.) Theorem 1.4.6. The Leech lattice is the unique densest lattice in E24 , up to scaling and isometries of E24 . We close this section with a short summary on the recent progress of L. Fejes T´oth’s [101] “sausage conjecture” that one can regard as an important basic problem on finite sphere packings closely related to the other questions of this section. According to this conjecture if in Ed , d 5 we take n 1 non-overlapping unit balls, then the volume of their convex hull is at least as large as the volume of the convex hull of the “sausage arrangement” of n non-overlapping unit balls under which we mean an arrangement whose centers lie on a line of Ed such that the unit balls of any two consecutive centers touch each other. By optimizing the methods developed by Betke, Henk, and Wills [25, 26], finally Betke and Henk [24] succeeded in proving the sausage conjecture of L. Fejes T´oth in any dimension of at least 42. Thus, we have the following theorem. Theorem 1.4.7. The sausage conjecture holds in Ed for all d 42. It remains an interesting challenge to prove or disprove the sausage conjecture of L. Fejes T´oth for the dimensions between 5 and 41. Conjecture 1.4.8. Let 5 d 41 be given. Then the volume of the convex hull of n 1 non-overlapping unit balls in Ed is at least as large as the volume of the convex hull of the “sausage arrangement” of n non-overlapping unit balls which is an arrangement whose centers lie on a line of Ed such that the unit balls of any two consecutive centers touch each other.

1.5 Uniformly Stable and Periodic Extreme Lattices The notion of solidity, introduced by L. Fejes T´oth [100] to overcome difficulties of the proper definition of density in the hyperbolic plane, has been proved very useful and stimulating. Roughly speaking, a family of convex sets generating a packing is said to be solid if no proper rearrangement of any finite subset of the packing elements can provide a packing. More concretely, a circle packing in the plane of constant curvature is called solid if no finite subset of the circles can be rearranged such that the rearranged circles together with the rest of the circles form a packing not congruent to the original. An (easy) example for solid circle packings is the family of incircles of a regular tiling fp; 3g for any p 3. In fact, a closer look at this example led L. Fejes T´oth [102] to the following simple sounding but difficult problem: he conjectured that the incircles of a regular tiling fp; 3g form a strongly solid packing for any p 5; that is, by removing any circle from the packing the

1.5 Uniformly Stable and Periodic Extreme Lattices

15

Fig. 1.2 Fejes T´oth conjecture: the incircles of f6; 3g form a strongly solid packing

remaining circles still form a solid packing. This conjecture has been verified for p D 5 by B¨or¨oczky [68] and Danzer [87] and for p 8 by A. Bezdek [27]. Thus, we have the following theorem. Theorem 1.5.1. The incircles of a regular tiling fp; 3g form a strongly solid packing for p D 5 and for any p 8. The outstanding open question left is the following (Fig. 1.2). Conjecture 1.5.2. The incircles of a regular tiling fp; 3g form a strongly solid packing for p D 6 as well as for p D 7. In connection with solidity and finite stability (of circle packings) the notion of uniform stability (of sphere packings) has been introduced by the author, A. Bezdek, and Connelly [56]. According to this a sphere packing (in the space of constant curvature) is said to be uniformly stable if there exists an " > 0 such that no finite subset of the balls of the packing can be rearranged such that each ball is moved by a distance less than " and the rearranged balls together with the rest of the balls form a packing different from the original one (not congruent to the original one). Now, suppose that P is a packing of (not necessarily) congruent balls in Ed . Let GP be the contact graph of P, where the centers of the balls serve as the vertices of GP and an edge is placed between two vertices when the corresponding two balls are tangent. The following basic principle can be used to show that many packings are uniformly stable. For the more technical definitions of “critical volume condition” and “infinitesimal rigidity” entering in the theorem below we refer the interested reader to the proof discussed in the proper section of this book.

16

1 Unit Sphere Packings

Theorem 1.5.3. Suppose that Ed ; d 2 can be tiled face-to-face by congruent copies of finitely many convex polytopes P1 ; P2 ; : : : ; Pm such that the vertices and edges of that tiling form the vertex and edge system of the contact graph GP of some ball packing P in Ed . Assume that each Pi and the graph GP restricted to the vertices of Pi (and regarded as a strut graph), satisfy the critical volume condition and assume that the bar framework G P (which is GP with all the struts changed to bars) restricted to the vertices of Pi is infinitesimally rigid. Then the packing P is uniformly stable. Actually, [56] proves a more general theorem (see Theorem 2.6.12 in this book), however, for the applications mentioned below the above version is sufficient. Namely, by taking a closer look at the Delaunay tilings of a number of lattice sphere packings one can derive the following corollary. The underlying symmetries of the lattices in question help a great deal to apply Theorem 1.5.3 in an efficient way (for more details see [56]). Corollary 1.5.4. The densest lattice sphere packings A2 ; A3 ; D4 ; D5 ; E6 ; E7 ; E8 (say, of unit balls) up to dimension 8 are all uniformly stable. In fact, the lattice unit sphere packings Ad and Dd (with minimum non-zero lattice vectors having length 2) are uniformly stable for all d 3. Recall that a packing of unit balls is called periodic if it is a union of finitely many translates of a lattice packing of unit balls. Perhaps a more striking version of Corollary 1.5.4, which can be derived from Theorem 1.5.3 is the following theorem: the lattice sphere packings of Corollary 1.5.4 are all periodic extreme, i.e., they cannot be locally modified to yield a periodic packing (of unit balls) with higher density. (For more details see Sects. 2.7 and 2.8 in [56].) This result was further strengthened by Sch¨urmann [169] proving that all perfect and strongly eutactic lattices (including the ones mentioned above and the Leech lattice) are periodic extreme. This naturally leads us to mention the following striking conjecture of Zassenhaus (see [169]). Conjecture 1.5.5. The largest density of unit ball packings in Ed ; d 4 can always be attained by periodic packings of unit balls. Last we mention another corollary of Theorem 1.5.3 (for details see [56]), which was also observed by B´ar´any and Dolbilin [19] and which supports the abovementioned conjecture of L. Fejes T´oth. Corollary 1.5.6. Consider the triangular packing of circular disks of equal radii in E2 where each disk is tangent to exactly six others. Remove one disk to obtain the packing P 0 . Then the packing P 0 is uniformly stable.

Chapter 2

Proofs on Unit Sphere Packings

Abstract The proofs presented in this chapter can be grouped as follows. We prove lower and upper estimates for the contact numbers of packings of n unit balls in Euclidean 3-space. One can regard this problem as a combinatorial relative of the Kepler problem on the densest unit sphere packings. Next, we give lower estimates for the surface volume of Voronoi cells in packings of unit balls in Euclidean d -space for all d 2 and then we improve those estimates in dimensions d 8. All these results imply upper bounds for the usual density of unit ball packings. Returning to the 3-dimensional Euclidean space we give lower bounds for the average surface area (resp., average edge curvature) of the cells in an arbitrary normal tiling with each cell holding a unit ball. On the one hand, it leads to a new version of the Kepler problem on unit sphere packings on the other hand, it generates a new relative of Kelvin’s foam problem. Finally, we find sufficient conditions for sphere packings being uniformly stable, a property that holds for all densest lattice sphere packings up to dimension 8.

2.1 Proof of Theorem 1.1.6

Theorem 1.1.6. 2

3 for all n 2; .i / C.n/ < 6n 0:926n p 3 2 2 .i i / Cf cc .n/ < 6n 3 18 n 3 D 6n 3:665 : : : n 3 for all n 2; p 2 2 .i i i / 6n 3 486n 3 < Cf cc .n/ C.n/ for all n D k.2k3 C1/ with k 2.

K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 2, © Springer International Publishing Switzerland 2013

17

18

2 Proofs on Unit Sphere Packings

Fig. 2.1 The isosceles triangle 4o1 pq

q ≈78° o1 α

α

p

2.1.1 An Upper Bound for Sphere Packings: Proof of (i) The proof presented in this section follows the proof of .i / of Theorem 1.1 in [54] and as such it is based on the recent breakthrough results of Hales [120]. The details are as follows. Let B denote the (closed) unit ball centered at the origin o of E3 and let P WD fc1 C B; c2 C B; : : : ; cn C Bg denote the packing of n unit balls with centers c1 ; c2 ; : : : ; cn in E3 having the largest number C.n/ of touching pairs among all packings of n unit balls in E3 . (P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.) Now, let rO WD 1:58731. The following statement shows the main property of rO that is needed for our proof of Theorem 1.1.6. Theorem 2.1.1. Let B1 ; B2 ; : : : ; B13 be 13 different members of a packing of unit balls in E3 . Assume that each ball of the family B2 ; B3 ; : : : ; B13 touches B1 . Let BO i be the closed ball concentric with Bi having radius rO , 1 i 13. Then the boundary bd.BO 1 / of BO 1 is covered by the balls BO 2 ; BO 3 ; : : : ; BO 13 , that is, O bd.BO 1 / [13 j D2 Bj : Proof. Let oi be the center of the unit ball Bi , 1 i 13 and assume that B1 is tangent to the unit balls B2 ; B3 ; : : : ; B13 at the points tj 2 bd.Bj / \ bd.B1 /; 2 j 13. Let ˛ denote the measure of the angles opposite to the equal sides of the isosceles triangle 4o1 pq with dist.o1 ; p/ D 2 and dist.p; q/ D dist.o1 ; q/ D rO , where dist.; / denotes the Euclidean distance between the corresponding two points. Clearly, cos ˛ D 1rO with ˛ < 3 (Fig. 2.1). Lemma 2.1.2. Let T be the convex hull of the points t2 ; t3 ; : : : ; t13 . Then the radius of the circumscribed circle of each face of the convex polyhedron T is less than sin ˛. Proof. Let F be an arbitrary face of T with vertices tj ; j 2 IF f2; 3; : : : ; 13g and let cF denote the center of the circumscribed circle of F . Clearly, the triangle 4o1 cF tj is a right triangle with a right angle at cF and with an acute angle of measure ˇF at o1 for all j 2 IF . We have to show that ˇF < ˛. We prove this by contradiction. Namely, assume that ˛ ˇF . Then either 3 < ˇF or ˛ ˇF 3 . First, let us take a closer look of the case 3 < ˇF . Reflect the point o1 about the plane of F and label the point obtained by o01 (Fig. 2.2).

2.1 Proof of Theorem 1.1.6 Fig. 2.2 The plane reflections to obtain o01 and o001

19 o1``

oj

o1`

tj

CF βF o1

Clearly, the triangle 4o1 o01 oj is a right triangle with a right angle at o01 and with an acute angle of measure ˇF at o1 for all j 2 IF . Then reflect the point o1 about o01 and label the point obtained by o001 furthermore, let B001 denote the unit ball centered at o001 . As 3 < ˇF therefore dist.o1 ; o001 / < 2 and so, one can simply translate B001 along the line o1 o001 away from o1 to a new position say, B000 1 such that it is tangent to B1 . However, this would mean that B1 is tangent to 13 non-overlapping unit balls namely, to B000 1 ; B2 ; B3 ; : : : ; B13 , clearly contradicting to the well-known fact [170] that this number cannot be larger than 12. Thus, we are left with the case when ˛ ˇF 3 . By repeating the definitions of o01 , o001 , and B001 , the inequality ˇF 3 implies in a straightforward way that the 14 unit balls B1 ; B001 ; B2 ; B3 ; : : : ; B13 form a packing in E3 . Moreover, the inequality ˛ ˇF yields that dist.o1 ; o001 / 4 cos ˛ D 4rO D 2:51998 : : : < 2:52. Finally, notice that the latter inequality contradicts to the following recent result of Hales [120]. Theorem 2.1.3. Let B1 ; B2 ; : : : ; B14 be 14 different members of a packing of unit balls in E3 . Assume that each ball of the family B2 ; B3 ; : : : ; B13 touches B1 . Then the distance between the centers of B1 and B14 is at least 2:52. This completes the proof of Lemma 2.1.2. Now, we are ready to prove Theorem 2.1.1. First, we note that by projecting the faces F of T from the center point o1 onto the sphere bd.BO 1 / we get a tiling of bd.BO 1 / into spherically convex polygons FO . Thus, it is sufficient to show that if F is an arbitrary face of T with vertices tj ; j 2 IF f2; 3; : : : ; 13g, then its central projection FO bd.BO 1 / is covered by the closed balls BO j ; j 2 IF f2; 3; : : : ; 13g. Second, in order to achieve this it is sufficient to prove that the projection cO F of the center cF of the circumscribed circle of F from the center point o1 onto the sphere bd.BO 1 / is covered by each of the closed balls BO j ; j 2 IF f2; 3; : : : ; 13g. Indeed, if in the triangle 4o1 oj cO F the measure of the angle at o1 is denoted by ˇF , then Lemma 2.1.2 implies in a straighforward way that ˇF < ˛. Hence, based on dist.o1 ; oj / D 2 and dist.o1 ; cO F / D rO , a simple comparison of the triangle 4o1 oj cO F with the triangle 4o1 pq yields that dist.oj ; cO F / < rO holds for all j 2 IF f2; 3; : : : ; 13g, finishing the proof of Theorem 2.1.1.

20

2 Proofs on Unit Sphere Packings

Fig. 2.3 Voronoi cells of a packing with yellow ci C B’s and blue ci C rO B’s

S Next, let us take the union niD1 .ci C rO B/ of the closed balls c1 C rO B; c2 C rO B; : : : ; cn C rB O of radii rO centered at the points c1 ; c2 ; : : : ; cn in E3 . Theorem 2.1.4.

vol3

nvol3 .B/ Sn < 0:7547; O i D1 .ci C rB/

where vol3 ./ refers to the 3-dimensional volume of the corresponding set. S Proof. First, partition niD1 .ci C rO B/ into truncated Voronoi cells as follows. Let Pi denote the Voronoi cell of the packing P assigned to ci C B, 1 i n, that is, let Pi stand for the set of points of E3 that are not farther away from ci than from any other cj with j ¤ i; 1 j n. Then, recall the well-known fact (see for example, [99]) that the Voronoi cells Pi , 1 i n just introduced form a tiling of E3 . Based on this it is easy to see that the truncated Voronoi O 1i n Sncells Pi \ .ci C rB/, generate a tiling of the non-convex container .c C r O B/ for the packing P. i i D1 p Second, as 2 < rO therefore the following very recent result of Hales [120] (see Lemma 9.13 on p. 228) applied to the truncated Voronoi cells Pi \ .ci C rO B/, 1 i n implies the inequality of Theorem 2.1.4 in a straightforward way (Fig. 2.3). Theorem 2.1.5. Let F be an arbitrary (finite or infinite) family of non-overlapping unit balls in E3 with the unit ball B centered at the origin o of E3 belonging to F . Let P stand for the Voronoi cell of the packing F assigned top B. Let Q denote a regular dodecahedron circumscribed B (having circumradius 3 tan 5 D 1:2584 : : :). p Finally, let r WD 2 D 1:4142 : : : and let rB denote the ball of radius r centered at the origin o of E3 . Then vol3 .B/ vol3 .B/ vol3 .B/ < 0:7547: vol3 .P/ vol3 .P \ rB/ vol3 .Q/

2.1 Proof of Theorem 1.1.6

21

This finishes the proof of Theorem 2.1.4. The well-known isoperimetric inequality [155] applied to

Sn

i D1

.ci C rO B/ yields

Lemma 2.1.6. 36vol23

n [

! .ci C rO B/ svol32 bd

i D1

n [

!! .ci C rB/ O

;

i D1

where svol2 ./ refers to the 2-dimensional surface volume of the corresponding set. Thus, Theorem 2.1.4 and Lemma 2.1.6 generate the following inequality. Corollary 2.1.7. 2

4

2

15:159805n 3 < 15:15980554 : : : n 3 D

< svol2 bd

n [

.0:7547/ !!

.ci C rO B/

2

2 3

n3

:

i D1

Now, assume that ci CB 2 P is tangent to cj CB 2 P for all j 2 Ti , where Ti f1; 2; : : : ; ng stands for the family of indices 1 j n for which dist.ci ; cj / D 2. Then let SOi WD bd.ci C rO B/ and let cO ij be the intersection of the line segment ci cj with SOi for all j 2 Ti . Moreover, let CSOi .Ocij ; 6 / (resp., CSOi .Ocij ; ˛/) denote the open spherical cap of SOi centered at cO ij 2 SOi having angular radius 6 (resp., ˛ with 0 < ˛ < 2 and cos ˛ D 1rO ). Clearly, the family fCSOi .Ocij ; 6 /; j 2 Ti g consists of pairwise disjoint open spherical caps of SOi ; moreover, P P cij ; 6 / j 2Ti svol2 CSOi .O j 2Ti Sarea C.uij ; 6 / ; D (2.1) Sarea [j 2Ti C.uij ; ˛/ svol2 [j 2Ti CSOi .Ocij ; ˛/ where uij WD 12 .cj ci / 2 S2 WD bd.B/ and C.uij ; 6 / S2 (resp., C.uij ; ˛/ S2 ) denotes the open spherical cap of S2 centered at uij having angular radius 6 (resp., ˛) and where Sarea./ refers to the spherical area measure on S2 . Now, Moln´ar’s density bound (Satz I in [148]) implies that Sarea C.uij ; 6 / < 0:89332 : Sarea [j 2Ti C.uij ; ˛/

P

j 2Ti

In order to estimate svol2 bd

n [ i D1

!! .ci C rO B/

(2.2)

22

2 Proofs on Unit Sphere Packings

from above let us assume that m members of P have 12 touching neighbours in P and k members of P have at most 9 touching neighbours in P. Thus, n m k members of P have either 10 or 11 touching neighbours in P. (Here we have used the well-known fact that 3 D 12, that is, no member of P can have more than 12 touching neighbours.) Without loss of generality we may assume that 4 k n m. p First, we note that Sarea C.uij ; 6 / D 2.1 cos 6 / D 2.1 23 / and p svol2 CSOi .Ocij ; 6 / D 2.1 23 /Or 2 . Second, recall Theorem 2.1.1 according to whichSif a member of P say, ci C B has exactly 12 touching neighbours in P, then SOi j 2Ti .cj C rO B/. These facts together with (2.1) and (2.2) imply the following estimate. S .nm k/C24:53902k : Corollary 2.1.8. svol2 bd niD1 .ci C rO B/ < 24:53902 3 Proof. svol2 bd p

10 2.1 23 /Or 2 < 4 rO 0:89332 2

!

n [

!! .ci C rO B/

i D1 p

3 2.1 23 /Or 2 .n m k/ C 4 rO 0:89332

!

2

< 7:91956.n m k/ C 24:53902k <

k

24:53902 .n m k/ C 24:53902k : 3

Hence, Corollaries 2.1.7 and 2.1.8 yield in a straightforward way that 2

1:85335n 3 3k < n m k :

(2.3)

Finally, as the number C.n/ of touching pairs in P is obviously at most 1 .12n .n m k/ 3k/ ; 2 therefore (2.3) implies that C.n/

1 2 2 .12n .n m k/ 3k/ < 6n 0:926675n 3 < 6n 0:926n 3 ; 2

finishing the proof of .i / in Theorem 1.1.6.

2.1 Proof of Theorem 1.1.6

23

2.1.2 An Upper Bound for the fcc Lattice: Proof of (ii) Although the idea of the proof of .i i / is similar to that of .i / they differ in the combinatorial counting part (see (2.9)) as well as in the density estimate for packings of spherical caps of angular radii 6 (see (2.8)). Moreover, the proof of .i i / is based p on the new parameter value rN WD 2 (replacing rO D 1:81383). The details are as follows. First, recall that if f cc denotes the face-centered cubic lattice with shortest nonzero lattice vector of length 2 in E3 and we place unit balls centered at each lattice point of f cc , then we get the fcc lattice packing of unit balls, labelled by Pf cc , in which each unit ball is touched by 12 others such that their centers form the vertices of a cuboctahedron. (Recall that a cuboctahedron is a convex polyhedron with 8 triangular faces and 6 square faces having 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron, i.e. an Archimedean solid, being vertex-transitive and edge-transitive.) Second, it is well known (see [99] for more details) that the Voronoi cell of each unitpball in Pf cc is a rhombicp dodecahedron (the dual of a cuboctahedron) of volume 32 (and of circumradius 2). Thus, the 4 density of Pf cc is p3 D p . 32

18

Now, let B denote the unit ball centered at the origin o 2 f cc of E3 and let P WD fc1 C B; c2 C B; : : : ; cn C Bg denote the packing of n unit balls with centers fc1 ; c2 ; : : : ; cn g f cc in E3 having the largest number Cf cc .n/ of touching pairs among all packings of n unit balls being a sub-packing of Pf cc . (P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.) As p the Voronoi cell of each unit ball in Pf cc is contained in a ball of radius rN D 2 therefore, based on the corresponding decomposition of S n N into truncated Voronoi cells, we get that i D1 .ci C rB/ nvol3 .B/ Sn < p D 0:7404 : : : : vol3 . i D1 .ci C rN B// 18

(2.4)

As a next step we apply the isoperimetric inequality [155]: 36vol23

n [ i D1

! .ci C rN B/

svol32

bd

n [

!! .ci C rB/ N

:

(2.5)

i D1

Thus, (2.4) and (2.5) yield in a straightforward way that n [ p 2 3 .ci C rN B/ 15:3532 : : : n D 4 18n 3 < svol2 bd 2 3

i D1

!! :

(2.6)

24

2 Proofs on Unit Sphere Packings

Now, assume that ci CB 2 P is tangent to cj CB 2 P for all j 2 Ti , where Ti f1; 2; : : : ; ng stands for the family of indices 1 j n for which dist.ci ; cj / D 2. Then let SNi WD bd.ci C rN B/ and let cN ij be the intersection of the line segment ci cj with SNi for all j 2 Ti . Moreover, let CSNi .Ncij ; 6 / (resp., CSNi .Ncij ; 4 /) denote the open spherical cap of SNi centered at cN ij 2 SNi having angular radius 6 (resp., 4 ). Clearly, the family fCSNi .Ncij ; 6 /; j 2 Ti g consists of pairwise disjoint open spherical caps of SNi ; moreover, P P cij ; 6 / j 2Ti svol2 CSNi .N j 2Ti Sarea C.uij ; 6 / D ; (2.7) svol2 [j 2Ti CSNi .Ncij ; 4 / Sarea [j 2Ti C.uij ; 4 / where uij D 12 .cj ci / 2 S2 and C.uij ; 6 / S2 (resp., C.uij ; 4 / S2 ) denotes the open spherical cap of S2 centered at uij having angular radius 6 (resp., 4 ). Now, the geometry of the cuboctahedron representing the 12 touching neighbours of an arbitrary unit ball in Pf cc implies in a straightforward way that p Sarea C.uij ; 6 / 3 6.1 / D 0:8038 : : : 2 Sarea [j 2Ti C.uij ; 4 /

P

j 2Ti

(2.8)

2 with equality when 12 spherical radius 6 are packed onS . caps of angular Finally, as svol2 C.uij ; 6 / D 2.1 cos 6 / and svol2 CSNi .Ncij ; 6 / D 2.1 p

3 2 /Nr 2

therefore (2.7) and (2.8) yield that

svol2 bd

n [ i D1

!! ci C rN B

p ! ! 3 2 p 2 2 1 4 rN 2 n rN Cf cc .n/ 3 2 6.1 2 / 1

D 8 n

4 Cf cc .n/ : 3

(2.9)

Thus, (2.6) and (2.9) imply that p 4 2 3 Cf cc .n/ : 4 18n 3 < 8 n 3 p 3

2

(2.10) 2

From (2.10) the inequality Cf cc .n/ < 6n 3 18 n 3 D 6n 3:665 : : : n 3 follows in a straightforward way for all n 2. This completes the proof of .i i / in Theorem 1.1.6.

2.1 Proof of Theorem 1.1.6

25

Fig. 2.4 The octahedral construction for k D 4

2.1.3 Octahedral Unit Sphere Packings: Proof of (iii) It is rather easy to show that for any positive integer k 2 there are n.k/ WD 2 2k 3 Ck D k.2k3 C1/ lattice points of the face-centered cubic lattice f cc such that 3 their convex hull is a regular octahedron K E3 of edge length 2.k 1/ having exactly k lattice points along each of its edges (see Fig. 2.4 for k D 4). Now, draw a unit ball around each lattice point of f cc \ K and label the packing of the n.k/ unit balls obtained in this way by Pf cc .k/. It is easy to check that if the center of a unit ball of Pf cc .k/ is a relative interior point of an edge (resp., of a face) of K, then the unit ball in question has 7 (resp., 9) touching neighbours in Pf cc .k/. Last but not least, any unit ball of Pf cc .k/ whose center is an interior pont of K has 12 touching neighbours in Pf cc .k/. Next we note that the number of lattice points of f cc lying in the relative interior of theedges (resp., faces) of K is 12.k 2/ D 12k 24 (resp., 8 12 .k 3/2 C 12 .k 3/ D 4.k 3/2 C 4.k 3/). Furthermore the number of lattice points of f cc in the interior of K is equal to 2 1 3 .k 2/ C .k 2/. Thus, the contact number C P .k/ of the packing Pf cc .k/ f cc 3 3 is equal to 12 2

7 2 9 24 1 3 .k 2/ C .k 2/ C 4.k 3/2 C 4.k 3/ C .12k 24/ C 3 3 2 2 2 D 4k 3 6k 2 C 2k:

As a result we get that C Pf cc .k/ D 6n.k/ 6k 2 :

(2.11)

26

2 Proofs on Unit Sphere Packings

p 3 2 Finally, as 2k3 < n.k/ therefore 6k 2 < 3 486n 3 .k/ and so, (2.11) implies .i i i / of Theorem 1.1.6 in a straightforward way.

2.2 Proof of Theorem 1.2.4 Theorem 1.2.4. The surface volume of any Voronoi cell in a packing of unit balls in d Ed ; d 2 is at least d! d .

2.2.1 The Voronoi Star of Voronoi Cells Without loss of generality we may assume that the d -dimensional unit ball B Ed centered at the origin o of Ed is one of the unit balls of the given unit ball packing in Ed ; d 2. Let V be the Voronoi cell assigned to B. We may assume that V is bounded; that is, V is a d -dimensional convex polytope in Ed . First, following [159], we dissect V into finitely many d -dimensional simplices as follows. Let Fi denote an arbitrary i -dimensional face of V, 0 i d 1. Let the chain F0 F1 Fd 1 be called a flag of V, and let F be the family of all flags of V. Now, let f 2 F be an arbitrary flag of V with the associated chain F0 F1 Fd 1 . Then let vi 2 Fd i be the point of Fd i closest to o, 1 i d . Finally, let Vf WD convfo; v1; : : : ; vd g, where conv./ stands for the convex hull of the given set. It is easy to see that the family V WD fVf j f 2 F and dim.Vf / D d g of d -dimensional simplices forms a tiling of V (i.e., [Vf 2V Vf D V and no two simplices of V have an interior point in common). This tiling is a rather special one, namely the d -dimensional simplices of V have o as a common vertex; moreover the union of their facets opposite to o is the boundary bdV of V. Finally, as shown in [159], for any Vf 2 V with Vf D convfo; v1; : : : ; vd g we have that r

2i kvi k D dist .o; convfvi ; vi C1 ; : : : ; vd g/ ; 1 i d; i C1

(2.12)

where dist.; / (resp., k k) stands for the Euclidean distance function (resp., norm) in Ed . Second, we define the Voronoi star V V assigned to the Voronoi cell V as follows. Definition 2.2.1. Let Vf 2 V with Vf D convfo; v1 ; : : : ; vd g. Then let v1 WD H \ linfv1 g, where H denotes the hyperplane parallel to the hyperplane afffv1 ; : : : ; vd g and tangent to B such that it separates o from afffv1 ; : : : ; vd g (with lin./ and aff./ standing for the linear and affine hulls of the given sets in Ed ). Finally, let Vf WD convfo; v1 ; v2 ; : : : ; vd g and let the Voronoi star V of V be defined as V WD [Vf 2V Vf .

2.2 Proof of Theorem 1.2.4

27

It follows from Definition 2.2.1 and from (2.12) that the following inequalities and (surface) volume formula hold: 1 kv1 k D dist o; convfv1 ; v2 ; : : : ; vd g kv1 k; r

2i kvi k D dist .o; convfvi ; vi C1 ; : : : ; vd g/ ; 2 i d; and i C1 vold .V / D

1 svold 1 .bdV/; d

(2.13)

(2.14)

(2.15)

where vold ./ (resp., svold 1 ./) refers to the d -dimensional (resp., .d 1/dimensional) volume (resp., surface volume) measure.

2.2.2 Estimating the Volume of a Voronoi Star from Below As an obvious corollary of (2.15), we find that Theorem 1.2.4 follows from the following theorem. Theorem 2.2.2. vold .V /

!d d

:

Proof. The main tool of our proof is the following lemma of Rogers. (See [159] and [160] for the original version of the lemma, which is somewhat different from the equivalent version below. Also, for a strengthening we refer the interested reader to Lemma 2.3.11.) Lemma 2.2.3. Let W WD convfo; w1 ; : : : ; wd g be a d -dimensional simplex of Ed having the property that linfwj wi j i < j d g is orthogonal to the vector wi in Ed for all 1 i d 1 (i.e., let W be a d -dimensional orthoscheme in Ed ). Moreover, let U WD convfo; u1 ; : : : ; ud g be a d -dimensional simplex of Ed such that kui k D dist .o; convfui ; ui C1 ; : : : ; ud g/ for all 1 i d . If kwi k kui k holds for all 1 i d , then vold .U/ vold .W/ ; vold .B \ W/ vold .B \ U/ where B stands for the d -dimensional unit ball centered at the origin o of Ed . Now, let q W be the orthoscheme of Lemma 2.2.3 with the additional property that

kwi k D

2i i C1

for all 1 i d . Notice that a regular d -dimensional simplex of

edge length 2 in Ed can be dissected into .d C 1/Š d -dimensional simplices, each congruent to W. This implies that

28

2 Proofs on Unit Sphere Packings

d D

vold .B \ W/ : vold .W/

(2.16)

Finally, let U WD Vf D convfo; v1 ; v2 ; : : : ; vd g for Vf 2 V. Clearly, (2.13) and (2.14) show that W and U, just introduced, satisfy the assumptions of Lemma 2.2.3. Thus, Lemma 2.2.3 and (2.16) imply that vold .Vf / 1 : d vold .B \ Vf /

(2.17)

Hence, (2.17) yields that X X vold .Vf / !d vold .B \ Vf / vold .Vf / D vold .V /; D d vol .B \ V / d f V 2V V 2V f

f

finishing the proof of Theorem 2.2.2.

2.3 Proof of Theorem 1.2.5 Theorem 1.2.5. The surface volume of any Voronoi cell in a packing of unit balls in d the d -dimensional Euclidean space Ed , d 8 is at least d! ; that is, the surface O d density of any unit sphere in its Voronoi cell in a unit sphere packing of Ed , d 8 is at most O d . Thus, the volume of any Voronoi cell in a packing of unit balls in Ed , d 8 is at least !O dd and so, the (upper) density of any unit ball packing in Ed , d 8 is at most O d (< d ).

2.3.1 Metric Properties of Voronoi Cells Let P be a bounded Voronoi cell, that is, a d -dimensional Voronoi polytope of a packing P of d -dimensional unit balls in Ed . Without loss of generality we may assume that the unit ball B D fx 2 Ed j dist.o; x/ D kxk 1g centered at the origin o of Ed is one of the unit balls of P with P as its Voronoi cell. Then P is the intersection of finitely many closed halfspaces of Ed each of which is bounded by a hyperplane that is the perpendicular bisector of a line segment ox with x being the center of some unit ball of P. Now, let Fd i be an arbitrary .d i /-dimensional face of P, 1 i d . Then clearly there are at least i C 1 Voronoi cells of P which meet along the face Fd i , that is, contain Fd i (one of which is, of course, P). Also, it is clear from the construction that the affine hull of centers of the unit balls sitting in all of these Voronoi cells is orthogonal to affFd i . Thus, there are unit balls of

2.3 Proof of Theorem 1.2.5

29

these Voronoi cells with centers fo; x1 ; : : : ; xi g such that X D convfo; x1 ; : : : ; xi g is an i -dimensional simplex and of course, affX is orthogonal to affFd i . Hence, if R.Fd i / denotes the radius of the .i 1/-dimensional sphere that passes through the vertices of X , then R.Fd i / D dist.o; affFd i /; where 1 i d: As the following statements are well known and their proofs are relatively straightforward, we refer the interested reader to the relevant section in [32] for the details of those proofs. p Lemma 2.3.1. If Fd i 1 Fd i and R.Fd i / D R < 2 for some i; 1 i d 1, then 2 p R.Fd i 1 /: 4 R2 q

R.Fd i / for all 1 i d . p Lemma 2.3.3. If R.Fd i / < 2 for some i; 1 i d , then the orthogonal projection of o onto affFd i belongs to relintFd i and so R.Fd i / D dist.o; Fd i /. Corollary 2.3.2.

2i i C1

2.3.2 Wedges of Types I, II, and III, and Truncated Wedges Let F0 F1 Fd 1 be an arbitrary flag of the Voronoi polytope P. Then let ri 2 Fd i be the uniquely determined point of the .d i /-dimensional face Fd i of P that is closest to the center point o of P; that is, let ri 2 Fd i such that kri k D minfkxk j x 2 Fd i g; where 1 i d: Definition 2.3.4. If the vectors r1 ; : : : ; ri are linearly independent in Ed , then we call convfo; r1 ; : : : ; ri g the i -dimensional Rogers simplex assigned to the subflag Fd i Fd 1 of the Voronoi polytope P, where 1 i d . If convfo; r1 ; : : : ; rd g Ed is the d -dimensional Rogers simplex assigned to the flag F0 Fd 1 of P, then convfrd i ; : : : ; rd g is called the i -dimensional base of the given d -dimensional Rogers simplex and dist.o; afffrd i ; : : : ; rd g/ D dist.o; affFi / D R.Fi / is called the height assigned to the i -dimensional base, where 1 i d . Definition 2.3.5. The i -dimensional simplex Y D convfo; y1; : : : ; yi g Ed with vertices y0 D o; y1 ; : : : ; yi is called an i -dimensional orthoscheme if for each j; 0 j i 1 the vector yj is orthogonal to the linear hull linfyk yj j j C 1 k i g, where 1 i d .

30

2 Proofs on Unit Sphere Packings

It is shown in [159] that the union of the d -dimensional Rogers simplices of the Voronoi polytope P is the polytope P itself and their interiors are pairwise disjoint. This fact together with Corollary 2.3.2 and Lemma 2.3.3 imply the following metric properties of Rogers simplices in a straightforward way. Lemma 2.3.6. (1) If convfo; r1 ; : : : ; ri g is an i -dimensional Rogers simplexq assigned to the subflag

Fd i Fd 1 of the Voronoi polytope P, then j2j krj k for all C1 1 j i , where 1 i d . (2) If pFd i Fd 1 is a subflag of the Voronoi polytope P with R.Fd i / < 2, then convfo; r1 ; : : : ; ri g is an i -dimensional Rogers simplex which is, in fact, an i -dimensional orthoscheme (in short, an i -dimensional Rogers orthoscheme) with the property that each rj 2 relintFd j ; 1 j i is the orthogonal projection of o onto affFd j , where 1 i d . (3) If F2 p Fd 1 is a subflag of the Voronoi polytope P Ed ; 3 d with R.F2 / < 2, then the union of the 2-dimensional bases of the d -dimensional Rogers simplices that contain the orthoscheme convfo; r1 ; : : : ; rd 2 g is the (uniquely determined) 2-dimensional face F2 of the Voronoi polytope P that is totally orthogonal to convfo; r1 ; : : : ; rd 2 g at the point rd 2 and so, krd 2 k D dist.o; affF2 / with rd 2 2 relintF2 .

Now we are ready for the definitions of wedges and truncated wedges. Recall that q for any 2-dimensional face F2 of the Voronoi polytope P Ed ; d 3 we have 2/ that 2.d d 1 R.F2 / (Fig. 2.5).

– B

x

y

rd-2

rd-1 F1

F2

rd WI ( resp.,WII )

Ed

Fig. 2.5 Wedges WI , WII , and truncated wedges WI , WII

o

2.3 Proof of Theorem 1.2.5

31

Definition 2.3.7. d (1) Let Fq 2 be a 2-dimensional face q of the Voronoi polytope P E ; d 3 2.d 2/ 2.d 1/ R.F2 / < and let convfo; r1 ; : : : ; rd 2 g be any with d 1 d .d 2/-dimensional Rogers simplex with rd 2 2 relintF2 . Then the union WI of the d -dimensional Rogers simplices of P that contain the orthoscheme convfo; r1 ; : : : ; rd 2 g is called a wedge of type I (generated by the .d 2/dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g). F2 is called the 2dimensional base of WI , and krd 2 k D dist.o; affF2 / is the height of WI assigned to the base F2 . d (2) Let Fq 2 be a 2-dimensional face qof the Voronoi polytope P E ; d 3 2.d 1/ 2d R.F2 / < with d d C1 and let convfo; r1 ; : : : ; rd 2 g be any .d 2/-dimensional Rogers simplex with rd 2 2 relintF2 . Then the union WII of the d -dimensional Rogers simplices of P that contain the orthoscheme convfo; r1 ; : : : ; rd 2 g is called a wedge of type II (generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g). F2 is called the 2dimensional base of WII , and krd 2 k D dist.o; affF2 / is the height of WII assigned to the base F2 . (3) Let convfo; r1 ; : : : ; rd g be the d -dimensional Rogers simplex assigned q to the

flag F0 F1 Fd 1 of the Voronoi polytope P Ed ; d 3 with R.F2 /. Then WIII D convfo; r1 ; : : : ; rd g is called a wedge of type III.

2d d C1

At this point, it useful q to recall, that for any vertex F0 of the Voronoi polytope d P E we have that d2d C1 R.F0 /. q o n . Definition 2.3.8. Let B D x 2 Ed j dist.o; x/ D kxk d2d C1 (1) If WI is a wedge of type I with the 2-dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 3, then WI D conv .B \ F2 / [ fo D r0 ; : : : ; rd 3 g is called the truncated wedge of type I with the 2-dimensional base B \ F2 generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g: (2) If WII is a wedge of type II with the 2-dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 3, then WII D conv .B \ F2 / [ fo D r0 ; : : : ; rd 3 g

32

2 Proofs on Unit Sphere Packings

is called the truncated wedge of type II with the 2-dimensional base B \ F2 generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g: As the following claim can be proved by Lemma 2.3.6 in a straightforward way, we leave the relevant details to the reader. Lemma 2.3.9. (1) Let WI (resp., WII ) denote the wedge of type I (resp., of type II) with the 2dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 3. If the points x; y 2 affF2 are chosen so that the triangle 4rd 2 xy has a right angle at the vertex x, then convfo; r1 ; : : : ; rd 2 ; x; yg is a d -dimensional orthoscheme. Moreover, if z 2 affF2 is an arbitrary point, then convfo D r0 ; : : : ; rd 3 ; zg is a .d 2/-dimensional orthoscheme. (2) Let WI denote the wedge of type I with the 2-dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo D r0 ; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 3. Let Q2 affF2 and Q2 affF2 be compact convex sets with relintQ2 \ relintQ2 D ;. If K2 D Q2 (resp., K2 D Q2 ) and Kj D conv.Kj 1 [ frd j g/ (resp., Kj D conv.Kj1 [ frd j g/) for j D 3; : : : ; d , then Kd D conv.Q2 [ fo D r0 ; : : : ; rd 3 g/ (resp., Kd D conv.Q2 [ fo D r0 ; : : : ; rd 3 g/), moreover relintKd \ relintKd D ;. A similar statement holds for WII . (3) Let WI (resp., WI ) denote the wedge of type I (resp., truncated wedge of type I) with the 2-dimensional base F2 (resp., B\F2 ) which is generated by the .d 2/dimensional Rogers orthoscheme convfo D r0 ; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 3. If K2 D F2 (resp., K2 D B \ F2 ) and Kj D conv.Kj 1 [ frd j g/ for j D 3; : : : ; d , then Kd D WI (resp., Kd D WI ). Similar statements hold for WII and WII . We close this section with the following important observation published in [32], and refer the interested reader to [32] for the details of the seven-page proof, which is based on Corollary 2.3.2 and Lemma 2.3.3. Lemma 2.3.10. Let B \ F2 be the 2-dimensional base of the type I truncated wedge WI (resp., type II truncated wedge WII ) in the Voronoi polytope P Ed of dimension d 8. Then the number of line segments of positive length in relbd.B \ F2 / is at most 4.

2.3.3 The Lemma of Comparison Recall that B D fx 2 Ed j dist.o; x/ D kxk 1g and let

2.3 Proof of Theorem 1.2.5

33

S D fx 2 Ed j dist.o; x/ D kxk D 1g: Then let H Ed be a hyperplane disjoint from the interior of the unit ball B and let Q H be an arbitrary .d 1/-dimensional compact convex set. If Œo; Q denotes the convex cone conv.fog [ Q/ with apex o and base Q, then the (volume) density ı.Œo; Q; B/ of the unit ball B in the cone Œo; Q is defined as ı.Œo; Q; B/ D

vold .Œo; Q \ B/ ; vold .Œo; Q/

where vold ./ refers to the corresponding d -dimensional Euclidean volume measure. It is natural to introduce the following very similar notion. The surface density O ı.Œo; Q; S / of the unit sphere S in the convex cone Œo; Q with apex o and base Q is defined by Svold 1 .Œo; Q \ S / O ı.Œo; Q; S / D ; vold 1 .Q/ where Svold 1 ./ refers to the corresponding .d 1/-dimensional spherical volume measure. O If h D dist.o; H /, then clearly h ı.Œo; Q; B/ D ı.Œo; Q; S /. We need the following statement, the first part of which is due to Rogers [159] and the second part of which has been proved by the author in [29]. Lemma 2.3.11. Let U D convfo; u1 ; : : : ; ud g be a d -dimensional orthoscheme in Ed and let V D convfo; v1 ; : : : ; vd g be a d -dimensional simplex of Ed such that kvi k D dist.o; convfvi ; vi C1 ; : : : ; vd g/ for all 1 i d 1. If 1 kui k kvi k holds for all 1 i d , then (1) ı.U; B/ ı.V; B/ and O O (2) ı.U; S / ı.V; S /. For the sake of completeness we mention the following statement that follows from Lemma 2.3.11 using the special decomposition of convex polytopes into Rogers simplices. Actually, the characterization of regular polytopes through the corresponding volume (resp., surface volume) inequality below was first observed by B¨or¨oczky and M´ath´en´e Bogn´ar [70] (resp., by the author [29]). (In fact, it is easy to see that the statement on surface volume implies the one on volume.) For more details on related problems we refer the interested reader to [69]. Corollary 2.3.12. Let U0 be a regular convex polytope in Ed with circumcenter o and let si denote the distance of an i -dimensional face of U0 from o, 0 i d 1. If V0 is an arbitrary convex polytope in Ed such that o 2 intV0 and the distance of any i -dimensional face of V0 from o is at least si for all 0 i d 1, then vold .V0 / vold .U0 / (resp., svold 1 .V0 / svold 1 .U0 /). Moreover, equality holds if and only if V0 is congruent to U0 and its circumcenter is o.

34

2 Proofs on Unit Sphere Packings

2.3.4 Volume Formulas for (Truncated) Wedges Definition 2.3.13. Let x1 ; : : : ; xn ; n 1 be points in Ed ; d 1 and let X Ed be an arbitrary convex set. If X0 D X and Xm D conv.fxn.m1/g [ Xm1 / for m D 1; : : : ; n, then we denote the final convex set Xn by Œx1 ; : : : ; xn ; X : Definition 2.3.14. Let WI (resp., WI ) denote the wedge (resp., truncated wedge) of type I with the 2-dimensional base F2 (resp., B \ F2 ) which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 4. Then let QI D Œr1 ; : : : ; rd 3 ; F2 resp:; QI D Œr1 ; : : : ; rd 3 ; B \ F2 be called the .d 1/-dimensional base of the type I wedge WI D Œo; QI (resp., type I truncated wedge WI D Œo; QI ). Similarly, we define the .d 1/dimensional bases QII and QII of WII and WII . Finally, let h1 D kr1 k; h2 D kr2 r1 k; : : : ; hd 2 D krd 2 rd 3 k: Lemma 2.3.15. Let WI (resp., WII ) denote the wedge of type I (resp., of type II ) with the 2-dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 4. Then we have the following volume formulas. Q d 2 2 (1) vold 1 .QI / D .d 1/Š h i D2 i vol2 .F2 / and Q d 2 vol2 .F2 /. (2) vold .WI / D d2Š h i i D1 Similar formulas hold for the corresponding dimensional volumes of QI , WI , QII , WII , QII , and WII . In general, if K affF2 is a convexdomain, then Qd 2 2 (3) vold 1 .Œr1 ; : : : ; rd 3 ; K/ D .d 1/Š i D2 hi vol2 .K/ and 2 Qd 2 (4) vold .Œo; r1 ; : : : ; rd 3 ; K/ D d Š i D1 hi vol2 .K/. Proof. The proof follows from Lemmas 2.3.6 and 2.3.9 in a straightforward way.

2.3.5 The Integral Representation of Surface Density The central notion of this section is the limiting surface density introduced as follows.

2.3 Proof of Theorem 1.2.5 Fig. 2.6 The limiting surface density ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S/

35

x aff F2

Tn(x)

rd-2

Ed

Definition 2.3.16. Let WI (resp., WII ) denote the wedge of type I (resp., of type II ) with the 2-dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 4. Then choose a coordinate system with two perpendicular axes in the plane affF2 meeting at the point rd 2 . Now, if x is an arbitrary point of the plane affF2 , then for a positive integer n let Tn .x/ affF2 denote the square centered at x having sides of length n1 parallel to the fixed coordinate axes. Then the limiting surface density ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / of the .d 1/-dimensional unit sphere S in the .d 2/-dimensional orthoscheme Œo; r1 ; : : : ; rd 3 ; x is defined by ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / D lim ıO .Œo; r1 ; : : : ; rd 3 ; Tn .x/; S / : n!1

Based on this we are able to give an integral representation of the surface density in a (truncated) wedge (Fig. 2.6). Lemma 2.3.17. Let WI (resp., WII ) denote the wedge of type I (resp., of type II ) with the 2-dimensional base F2 which is generated by the .d 2/dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 4. (1) If x 2 affF2 and y 2 affF2 are points such that kxk kyk, then ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / ıOlim .Œo; r1 ; : : : ; rd 3 ; y; S / : (2) For the surface densities of the unit sphere S in the wedge WI and in the truncated wedge WI we have the following formulas. 1 O I ; S /D Svold 1 .Œo; QI \ S / D ı.W vold 1 .QI / vol2 .F2 /

Z

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x F2

36

2 Proofs on Unit Sphere Packings

and O I ; S / D Svold 1 .Œo; QI \ S / ı.W vold 1 .QI /

D

Z

1 vol2 .B \ F2 /

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x; B\F2

where d x stands for the Euclidean area element in the plane affF2 . Similar formulas hold for WII and WII . (3) In general, if K affF2 is a convex domain, then the surface density of the unit sphere S in the d -dimensional convex cone Œo; r1 ; : : : ; rd 3 ; K with apex o and .d 1/-dimensional base Œr1 ; : : : ; rd 3 ; K can be computed as follows. O ı.Œo; r1 ; : : : ; rd 3 ; K; S / D

1 vol2 .K/

Z

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x: K

Proof.(1) It is sufficient to look at the case kxk < kyk. (The case kxk D kyk follows from this by standard limit procedure.) Then recall that ıO .Œo; r1 ; : : : ; rd 3 ; Tn .x/; S / D h1 ı .Œo; r1 ; : : : ; rd 3 ; Tn .x/; S / and ıO .Œo; r1 ; : : : ; rd 3 ; Tn .y/; S / D h1 ı .Œo; r1 ; : : : ; rd 3 ; Tn .y/; S / : Thus, it is sufficient to show that if n is sufficiently large, then ı .Œo; r1 ; : : : ; rd 3 ; Tn .x/; S / ı .Œo; r1 ; : : : ; rd 3 ; Tn .y/; S / : This we can get as follows. We can approximate the d -dimensional convex cone Œo; r1 ; : : : ; rd 3 ; Tn .x/ (resp., Œo; r1 ; : : : ; rd 3 ; Tn .y/) arbitrarily close with a finite (but possibly large) number of non-overlapping d -dimensional orthoschemes each containing the .d 3/-dimensional orthoscheme Œo; r1 ; : : : ; rd 3 as a face and each having all the edge lengths of the 3 edges going out from the vertex o and not lying on the face Œo; r1 ; : : : ; rd 3 close to kxk (resp., kyk) for n sufficiently large (see also Lemma 2.3.9). Thus, the claim follows from (1) of Lemma 2.3.11 rather easily. (2),(3) It is sufficient to prove the corresponding formula for K. A typical term of the Riemann–Lebesgue sum of 1 vol2 .K/

Z

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x K

2.3 Proof of Theorem 1.2.5

37

is equal to 1 ıO .Œo; r1 ; : : : ; rd 3 ; Tn .xm /; S / vol2 .Tn .xm //; m 2 M: vol2 .K/ Using Lemma 2.3.15 this turns out to be equal to vold 1 .Œr1 ; : : : ; rd 3 ; Tn .xm // O ı .Œo; r1 ; : : : ; rd 3 ; Tn .xm /; S / vold 1 .Œr1 ; : : : ; rd 3 ; K/ D

Svold 1 .Œo; r1 ; : : : ; rd 3 ; Tn .xm / \ S / : vold 1 .Œr1 ; : : : ; rd 3 ; K/

Finally, as the union of the non-overlapping squares Tn .xm /; m 2 M is a good approximation of the convex domain K in the plane affF2 we get that X Svold 1 .Œo; r1 ; : : : ; rd 3 ; Tn .xm / \ S / vold 1 .Œr1 ; : : : ; rd 3 ; K/ m2M P Svold 1 .Œo; r1 ; : : : ; rd 3 ; Tn .xm / \ S / D m2M vold 1 .Œr1 ; : : : ; rd 3 ; K/ is a good approximation of Svold 1 .Œo; r1 ; : : : ; rd 3 ; K \ S / O D ı.Œo; r1 ; : : : ; rd 3 ; K; S /: vold 1 .Œr1 ; : : : ; rd 3 ; K/ This completes the proof of Lemma 2.3.17.

2.3.6 Truncation of Wedges Increases the Surface Density Lemma 2.3.18. Let WI (resp., WII ) denote the wedge of type I (resp., of type II ) with the 2-dimensional base F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 4. Then O I ; S / ı.W O I ; S / resp:; ı.W O II ; S / ı.W O II ; S / : ı.W Proof. Notice that (1) of Lemma q 2.3.17 easily implies that if 0 < vol2 .F2 n B/, then

for any x 2 F2 with kx k D

2d d C1

we have that

38

2 Proofs on Unit Sphere Packings

Z

1 vol2 .F2 n B/

F2 nB

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x

ıOlim Œo; r1 ; : : : ; rd 3 ; x ; S Z 1 ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x: vol2 .B \ F2 / B\F2 Thus, if 0 < vol2 .F2 n B/, then (2) of Lemma 2.3.17 yields that O I ; S/ D ı.W

1 vol2 .F2 /

Z

vol2 .B \ F2 / 1 D vol2 .F2 / vol2 .B \ F2 / C

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x

F2

1 vol2 .F2 n B/ vol2 .F2 / vol2 .F2 n B/

1 vol2 .B \ F2 / vol2 .F2 / vol2 .B \ F2 /

Z

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x B\F2

Z

F2 nB

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x

Z

ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x B\F2

Z 1 vol2 .F2 n B/ ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x C vol2 .F2 / vol2 .B \ F2 / B\F2 Z 1 O I ; S /: D ıOlim .Œo; r1 ; : : : ; rd 3 ; x; S / d x D ı.W vol2 .B \ F2 / B\F2 As the same method works for WII and WII this completes the proof of Lemma 2.3.18.

2.3.7 Maximum Surface Density in Truncated Wedges of Type I Let WI denote the truncated wedge of type I with the 2-dimensional base B \ F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 8. By assumption F2 is a 2-dimensional face of the Voronoi polytope P with r

2.d 2/ h D R.F2 / < d 1

r

2.d 1/ : d

Let G0 affF2 (resp., G affF2 ) denote the closed circular disk of radius

2.3 Proof of Theorem 1.2.5

39

s g0 .h/ D

2 h2 2d 2 h resp:; g.h/ D p d C1 4 h2

q 2/ centered at the point rd 2 . It is easy to see that G relintG0 for all 2.d d 1 q q 2.d 1/ 2.d 1/ . (Moreover G D G0 for h D .) Notice that G0 D B \ h < d d affF2 , thus Corollary 2.3.2 implies that there is no vertex of pthe face F2 belonging to the relative interior of G0 . Moreover, as h D R.F2 / < 2 Lemma 2.3.1 yields that p 2 2 R.F1 / holds for any side F1 of the face F2 , hence G F2 and of 4h

course, G B \ F2 D G0 \ F2 . Now, let M affF2 be a square circumscribed 0 .h/ about G. A straightforward computation yields that gg.h/ is a strictly decreasing q q 2.d 2/ 2.d 1/ g0 .h/ d < 0 on the interval (i.e., dh function on the interval d 1 ; d g.h/ q q 2.d 2/ ; 2.dd1/ ) and d 1 q g0 q g

2.d 2/ d 1

2.d 2/ d 1

D

s

p 2d < 2: d C1

Thus, the vertices of the square M do not belong to G0 . Finally, as d 8 Lemma 2.3.10 implies that there are at most four sides of the face F2 that intersect the relative interior of G0 . The following statement is rather natural from the point of view of the local geometry introduced above, however, its three-page proof based on Lemmas 2.3.11 and 2.3.17 published in [32] is a bit technical and so, for that reason we do not prove it here; instead we refer the interested reader to the proper section in [32] (Fig. 2.7).

N aff F2

M G F2

Fig. 2.7 Surface density bounded from above by O ı.Œo; r1 ; : : : ; rd 3 ; G0 \ M ; S/

rd-2

G0

M0

40

2 Proofs on Unit Sphere Packings

Lemma 2.3.19. Let WI denote the truncated wedge of type I with the 2-dimensional base B \ F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 8. Then O O I ; S / ı.Œo; r1 ; : : : ; rd 3 ; G0 \ M ; S /: ı.W O It is clear from the construction that we can write ı.Œo; r1 ; : : : ; rd 3 ; G0 \ M ; S / as a function of d 2 variables, namely O O 1 ; : : : ; d 3 ; d 2 / D ı.Œo; r1 ; : : : ; rd 3 ; G0 \ M ; S /; . where 1 D kr1 k; : : : ; d 3 D krd 3 k; d 2 D krd 2 k D h. Corollary 2.3.2 and the assumption on h imply that r m1 D 1 1 ; : : : ; mi D r md 2 D

2i i ; : : : ; md 3 D i C1

2.d 2/ d 2 D h < d 1

r

r

2.d 3/ d 3 ; d 2

2.d 1/ : d

Notice that if kri k D mi for all 1 i d 2, then Œo; r1 ; : : : ; rd 3 ; G0 \ M can be dissected into four pieces each being congruent to W and therefore O ı.Œo; r1 ; : : : ; rd 3 ; G0 \ M ; S / D O d . Lemma 2.3.20. O 1 ; : : : ; d 3 ; d 2 / .m O 1 ; : : : ; md 3 ; md 2 / D O d : . Proof. For any fixed d 2 D h, (2) of Lemma 2.3.11 easily implies that O 1 ; : : : ; d 3 ; h/ .m O 1 ; : : : ; md 3 ; h/: . Finally, using Lemma 2.3.11 again, it is rather straightforward to show that the O 1 ; : : : ; md 3 ; h/ as a function of h is decreasing on the interval function .m q q 2.d 2/ 2.d 1/ . From this it follows that ; d 1 d O 1 ; : : : ; md 3 ; h/ .m O 1 ; : : : ; md 3 ; md 2 / D O d ; .m finishing the proof of Lemma 2.3.20. Thus, Lemmas 2.3.19 and 2.3.20 yield the following immediate estimate.

2.3 Proof of Theorem 1.2.5

41

Corollary 2.3.21. Let WI denote the truncated wedge of type I with the 2-dimensional base B \ F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 8. Then O I ; S / O d : ı.W

2.3.8 Surface Density in Truncated Wedges of Type II It is sufficient to prove the following statement. Lemma 2.3.22. Let WII denote the truncated wedge of type II with the 2-dimensional base B \ F2 which is generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 4. Then O II ; S / O d : ı.W Proof. By assumption F2 is a 2-dimensional face of the Voronoi polytope P with r

2.d 1/ h D R.F2 / < d

s 2d : d C1

q 2d Let G0 affF2 denote the closed circular disk of radius g0 .h/ D h2 d C1 p centered at the point rd 2 . As h D R.F2 / < 2, therefore Lemma 2.3.1 yields that s 2 2d p R.F1 / d C1 4 h2 holds for any side F1 of the face F2 . Thus, B \ F2 D G0 and so O II ; S / D ı.Œo; O ı.W r1 ; : : : ; rd 3 ; G0 ; S /: O It is clear from the construction that we can write ı.Œo; r1 ; : : : ; rd 3 ; G0 ; S / as a function of d 2 variables, namely O O . 1 ; : : : ; d 3 ; d 2 / D ı.Œo; r1 ; : : : ; rd 3 ; G0 ; S /;

42

2 Proofs on Unit Sphere Packings

where 1 D kr1 k; : : : ; d 3 D krd 3 k; d 2 D krd 2 k D h. Corollary 2.3.2 and the assumption on h imply that r m1 D 1 1 ; : : : ; mi D r md 2 D

2i i ; : : : ; md 3 D i C1

2.d 1/ d 2 D h < d

r

2.d 3/ d 3 ; d 2

s 2d : d C1

For any fixed d 2 D h, (2) of Lemma 2.3.11 easily implies that O . 1 ; : : : ; d 3 ; h/ O .m1 ; : : : ; md 3 ; h/: Finally, again applying (2) of Lemma 2.3.11 we immediately get that O .m1 ; : : : ; md 3 ; h/ O .m1 ; : : : ; md 3 ; md 2 / O d : This completes the proof of Lemma 2.3.22.

2.3.9 The Overall Estimate of Surface Density in Voronoi Cells Let P be a d -dimensional Voronoi polytope of a packing P of d -dimensional unit balls in Ed ; d 8. Without loss of generality we may assume that the unit ball B D fx 2 Ed j dist.o; x/ D kxk 1g centered at the origin o of Ed is one of the unit balls of P with P as its Voronoi cell. As before, let S denote the boundary of B. First, we dissect P into d -dimensional Rogers simplices. Then let convfo; r1 ; : : : ; rd g be one of these d -dimensional Rogers simplices assigned to the flag say, F0 Fd 1 of P. As ri 2 Fd i ; 1 i d it is clear that afffrd 2 ; rd 1 ; rd g D affF2 and so dist.o; afffrd 2 ; rd 1 ; rd g/ D dist.o; affF2 / D R.F2 /: q 2/ Notice that Corollary 2.3.2 implies that 2.d d 1 R.F2 /. Second, we group the d -dimensional Rogers simplices of P as follows. q q 2/ 2.d 1/ 1. If 2.d R.F / < , then we assign the Rogers simplex convfo; r1 ; 2 d 1 d : : : ; rd g to the type I wedge WI with the 2-dimensional base F2 generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 8.

2.4 Proof of Theorem 1.3.3

43

q q 2. If 2.dd1/ R.F2 / < d2d , then we assign the Rogers simplex convfo; r1 ; C1 : : : ; rd g to the type II wedge WII with the 2-dimensional base F2 generated by the .d 2/-dimensional Rogers orthoscheme convfo; r1 ; : : : ; rd 2 g of the Voronoi polytope P Ed ; d 8. q 3. If d2d R.F2 /, then we assign the Rogers simplex convfo; r1 ; : : : ; rd g to C1 itself as the type III wedge WIII .

As the wedges of types I–III of the given Voronoi polytope P sit over the 2-skeleton of P and form a tiling of P it is clear that each d -dimensional Rogers simplex of P belongs to exactly one of them. As a result, in order to show that d!d O S / D Svold 1 .S / D the surface density ı.P; svold 1 .bdP/ svold 1 .bdP/ of the unit sphere S in the Voronoi polytope P is bounded from above by O d , it is sufficient to prove the following inequalities. O .1/: O .2/: O .3/:

O I ; S / O d , ı.W O II ; S / O d , ı.W O III ; S / O d . ı.W

This final task is now easy. Namely, Lemma 2.3.18, Corollary 2.3.21, and O and .2/ O in a straightforward way. Finally, .3/ O follows with Lemma 2.3.22 yield .1/ the help of (2) of Lemma 2.3.11 rather easily. For the details of the proof of O d < d , based on the so-called “Lemma of Strict Comparison”, we refer the interested reader to the proper section in [32]. This completes the proof of Theorem 1.2.5.

2.4 Proof of Theorem 1.3.3 Theorem 1.3.3. Let T be an arbitrary normal tiling of E3 . Then the average surface 24 24 area of the cells in T is always at least p , i.e., s.T / p D 13:8564 : : :. 3 3

2.4.1 Average Surface Area of Cells in Normal Tilings of a Cube First, we prove the following “compact” version of Theorem 1.3.3. Theorem 2.4.1. If the cube C is partitioned into the convex cells Q1 ; Q2 ; : : : ; Qn each containing a unit ball E3 , then the sum of the surface areas of the n convex Pin n 24 24 cells is at least p3 n, i.e., i D1 sarea.Qi / p n. 3 Proof. It is well known that the Brunn-Minkowski inequality implies the following inequality:

44

2 Proofs on Unit Sphere Packings

sarea2 .Qi / 3vol.Qi /ecurv.Qi / ;

(2.18)

where 1 i n. (For a proof we refer the interested reader to p. 287 in [99].) In what follows it is more proper to use the inner dihedral angles ˇe WD ˛e and the relevant formula for the edge curvature: ecurv.Qi / D

X

L.e/ cot

e2E.Qi /

ˇe : 2

(2.19)

As, by assumption, Qi contains a unit ball therefore vol.Qi /

1 sarea.Qi / : 3

(2.20)

Hence, (2.18)–(2.20) imply in a straightforward way that sarea.Qi /

X

L.e/ cot

e2E.Qi /

ˇe 2

(2.21)

holds for all 1 i n. Now, let s C be a closed line segment along which exactly k members of the family fQ1 ; Q2 ; : : : ; Qn g meet having inner dihedral angles ˇ1 ; ˇ2 ; : : : ; ˇk . There are the following three possibilities: (a) s is on an edge of the cube C; (b) s is in the relative interior either of a face of C or of a face of a convex cell in the family fQ1 ; Q2 ; : : : ; Qn g; (c) s is in the interior of C and not in the relative interior of any face of any convex cell in the family fQ1 ; Q2 ; : : : ; Qn g. In each of the above cases we can make the following easy observations: (a) ˇ1 C ˇ2 C C ˇk D 2 with k 1; (b) ˇ1 C ˇ2 C C ˇk D with k 2; (c) ˇ1 C ˇ2 C C ˇk D 2 with k 3. As y D cot x is convex and decreasing over the interval 0 < x following inequalities must hold:

2

therefore the

k; (a) cot ˇ21 C cot ˇ22 C C cot ˇ2k k cot 4k ˇ1 ˇ2 ˇk (b) cot 2 C cot 2 C C cot 2 k cot 2k k; (c) cot ˇ21 C cot ˇ22 C C cot ˇ2k k cot k p1 k. 3

In short, the following inequality holds in all three cases: cot

ˇ2 ˇk 1 ˇ1 C cot C C cot p k: 2 2 2 3

(2.22)

2.4 Proof of Theorem 1.3.3

45

Thus, by adding together the inequalities (2.21) for all 1 i n and using (2.22) we get that n X

n 1 X X sarea.Qi / p L.e/ : 3 i D1 e2E.Qi / i D1

(2.23)

Finally, recall the elegant theorem of Besicovitch and Eggleston [23] claiming that the total edge length of any convex polyhedron containing a unit ball in E3 is always at least as large as the total edge length of a cube circumscribed a unit ball. This implies that X

L.e/ 24

(2.24)

e2E.Qi /

holds for all 1 i n. Hence, (2.23) and (2.24) finish the proof of Theorem 2.4.1.

2.4.2 Average Surface Area of Cells in Normal Tilings Second, we take a closer look of the given normal tiling T and using the above proof of Theorem 2.4.1 we give a proof of Theorem 1.3.3. The details are as follows. By assumption D WD supfdiam.Pi /ji D 1; 2; : : : g < 1. Thus, clearly each closed ball of radius D in E3 contains at least one of the convex polyhedra Pi ; i D 1; 2; : : : (forming the tiling T of E3 ). Now, let CLN ; N D 1; 2; : : : be an arbitrary sequence of cubes centered at the origin o with edges parallel to the coordinate axes of E3 and having edge length LN ; N D 1; 2; : : : with limN !1 Ln D 1. It follows that CLN g < 1: vol.CLN /

(2.25)

vol.CLN C2D / vol.CLN 2D / vol.CLN / cardfi jPi \ bdCLN ¤ ;g cardfi jBi CLN g vol.CLN / 4 3 cardfi jBi CLN g

(2.26)

0 < lim inf

4 3 cardfi jBi

N !1

CLN g lim sup vol.CLN / N !1

4 3 cardfi jBi

Note that clearly

moreover, vol.CLN C2D / vol.CLN 2D / D 0: N !1 vol.CLN / lim

(2.27)

46

2 Proofs on Unit Sphere Packings

Thus, (2.25)–(2.27) imply in a straightforward way that cardfi jPi \ bdCLN ¤ ;g D0: N !1 cardfi jBi CLN g

(2.28)

lim

Moreover, (2.21) yields that X

sarea.Pi / ecurv.Pi / D

L.e/ cot

e2E.Pi /

ˇe 2

(2.29)

holds for all i D 1; 2; : : : . As a next step, using sarea.Pi / D sarea .bd.Pi \ CL / n bdCL / C ıi

(2.30)

and X

ecurv.Pi /

L.e/ cot

e2E.bd.Pi \CL /nbdCL /

ˇe 2

(2.31)

(with bd./ denoting the boundary of the corresponding set) we obtain the following from (2.29): X

sarea .bd.Pi \ CL / n bdCL / C ıi

L.e/ cot

e2E.bd.Pi \CL /nbdCL /

ˇe ; 2

(2.32)

where clearly 0 ıi sarea.Pi /. Hence, (2.32) combined with (2.22) yields Corollary 2.4.2. f .L/ WD

X

X

sarea bd.Pi \ CL / n bdCL C

fi jintPi \CL ¤;g

X 1 g.L/ WD p 3 fi jintP \C i

L ¤;g

ıi

fi jPi \bdCL ¤;g

X

L.e/

e2E.bd.Pi \CL /nbdCL /

Now, it is easy to see that f .L/ D .L/ C

X

sarea.Pi \ CL / ;

fi jBi CL g

where 0 .L/ 2

P fi jPi \bdCL ¤;g

sarea.Pi /.

(2.33)

2.4 Proof of Theorem 1.3.3

47

Moreover, (2.24) implies that X

g.L/ .L/ C

fi jBi CL

where 0 .L/

P

P fi jPi \bdCL ¤;g

e2E.Pi /

24 p ; 3 g

(2.34)

L.e/.

Lemma 2.4.3. A WD supfsarea.Pi /ji D 1; 2; : : : g < 1 and E WD supf

X

L.e/ji D 1; 2; : : : g < 1 :

e2E.Pi /

Proof. As D D supfdiam.Pi /ji D 1; 2; : : : g < 1 therefore according to Jung’s q 3 theorem [90] each Pi is contained in a closed ball of radius 8 D in E3 . Thus, A 32 D 2 < 1. For a proof of the other claim recall that Pi contains the unit ball Bi centered at oi . If the number of faces of Pi is fi , then Pi must have at least fi neighbours (i.e., cells of T that have at least one point in common with Pi ) and as each neighbour is contained in the closed 3-dimensional ball of radius 2D centered at oi therefore the number of neighbours of Pi is at most .2D/3 1 and so, fi 8D 3 1. (Here, we have used the fact that each neighbour contains a unit ball and therefore its volume is larger than 4 3 .) Finally, Euler’s formula implies that the number of edges of Pi is at most 3fi 6 24D 3 9. Thus, E 24D 4 9D < 1 (because the length of any edge of Pi is at most D). Thus, Corollary 2.4.2, (2.33), (2.34), and Lemma 2.4.3 imply the following inequality in a straightforward way. Corollary 2.4.4. 2Acardfi jPi \ bdCL ¤ ;g C

P fi jBi CL g

cardfi jBi CL g

Ecardfi jPi \ bdCL ¤ ;g C

sarea.Pi \ CL /

P fi jBi CL g

24 p 3

cardfi jBi CL g

Finally, Corollary 2.4.4 and (2.28) yield that P lim inf N !1

fi jBi CLN g

sarea.Pi \ CLN /

cardfi jBi CLN g

finishing the proof of Theorem 1.3.3.

24 p ; 3

:

48

2 Proofs on Unit Sphere Packings

2.5 Proof of Theorem 1.3.5 Theorem 1.3.5. Let T be an arbitrary normal tiling of E3 . Then s.T / ec.T / 24 p D 13:8564 : : :. Moreover, if T is a Voronoi tiling of a unit ball packing in E3 , 3

then ec.T /

2 2 p p 6 6 arcsin p1 6

D 14:6176 : : : .

3

2.5.1 Average Edge Curvature of Cells in Normal Tilings If T is an arbitrary normal tiling of E3 with the underlying packing P of the unit balls Bi ; i D 1; 2; : : : , then (2.21) implies sarea.Pi \ CL / ecurv.Pi \ CL / and therefore s.T / ec.T / follows in a straightforward way. So, we are left to 24 show that ec.T / p . In order to achieve that, we take a closer look of the given 3 normal tiling T and using some of the properly modified estimates of the proof of 24 . The details are as follows. Theorem 1.3.3 we derive the inequality ec.T / p 3 We start with the following immediate analogue of (2.32): X

ecurv.Pi / D ıi C

L.e/ cot

e2E.bd.Pi \CL /nbdCL /

ˇe ; 2

(2.35)

where 0 ıi ecurv.Pi / sarea.Pi /. If X

ecurv bd.Pi \ CL / n bdCL WD

L.e/ cot

e2E.bd.Pi \CL /nbdCL /

ˇe ; 2

then (2.35) combined with (2.22) yields Corollary 2.5.1. f .L/ WD

X

X

ecurv bd.Pi \ CL / n bdCL C

fi jintPi \CL ¤;g

X 1 g.L/ D p 3 fi jintP \C i

L ¤;g

fi jPi \bdCL ¤;g

X e2E.bd.Pi \CL /nbdCL /

L.e/

ıi

2.5 Proof of Theorem 1.3.5

49

Now, it is easy to see that f .L/ D .L/ C

X

ecurv.Pi \ CL / ;

(2.36)

fi jBi CL g

P where 0 .L/ 2 fi jPi \bdCL ¤;g ecurv.Pi /. Moreover, g.L/ must satisfy (2.34). Thus, Corollary 2.5.1, (2.36), (2.34), and Lemma 2.4.3 imply the following inequality in a straightforward way. Corollary 2.5.2. 2Acardfi jPi \ bdCL ¤ ;g C

P fi jBi CL g

cardfi jBi CL g

Ecardfi jPi \ bdCL ¤ ;g C

ecurv.Pi \ CL /

P fi jBi CL g

24 p 3

cardfi jBi CL g

:

Hence, Corollary 2.5.2 and (2.28) yield that P lim inf N !1

fi jBi CLN g

ecurv.Pi \ CLN /

cardfi jBi CLN g

finishing the proof of the inequality ec.T /

24 p ; 3

24 p . 3

2.5.2 Average Edge Curvature of Voronoi Cells in Unit Ball Packings Finally, let T be the Voronoi tiling of a unit ball packing in E3 consisting of the Voronoi cells Pi ; i D 1; 2; : : : (each containing a unit ball). First, recall the inequality ecurv.Pi / > 2 mwidth.Pi / ;

(2.37)

where i D 1; 2; : : : and mwidth./ denotes the mean width of the corresponding set. (For more details on this inequality see p. 287 in [99] as well as the relevant discussion on p. 392 in [31].) Second, recall that according to a recent result of the author [31] the inequality mwidth.Pi / p p D 2:3264 : : : 6 6 arcsin p13 6

(2.38)

50

2 Proofs on Unit Sphere Packings

holds for all i D 1; 2; : : : . Thus, (2.37) and (2.38) yield 2 2 ecurv.Pi / > p p D 14:6176 : : : 6 6 arcsin p13 6 from which it follows in a straightforward way that ec.T / 14:6176 : : : , finishing the proof of Theorem 1.3.5.

2.6 Proof of Theorem 1.5.3 Theorem 1.5.3. Suppose that Ed ; d 2 can be tiled face-to-face by congruent copies of finitely many convex polytopes P1 ; P2 ; : : : ; Pm such that the vertices and edges of that tiling form the vertex and edge system of the contact graph GP of some ball packing P in Ed . Assume that each Pi and the graph GP restricted to the vertices of Pi (and regarded as a strut graph), satisfy the critical volume condition and assume that the bar framework G P (which is GP with all the struts changed to bars) restricted to the vertices of Pi is infinitesimally rigid. Then the packing P is uniformly stable.

2.6.1 The Signed Volume of Convex Polytopes Definition 2.6.1. Let P WD convfp1 ; p2 ; : : : ; pn g be a d -dimensional convex polytope in Ed ; d 2 with vertices p1 ; p2 ; : : : ; pn . If F WD convfpi1 ; : : : ; pik g is an arbitrary face of P, then the barycenter of F is cF WD

k 1X pi : k j D1 j

(2.39)

Let F0 F1 Fl ; 0 l d 1 denote a sequence of faces, called a (partial) flag of P, where F0 is a vertex and Fi 1 is a facet (a face one dimension lower) of Fi for i D 1; : : : ; l. Then the simplices of the form convfcF0 ; cF1 ; : : : ; cFl g constitute a simplicial complex CP whose underlying space is the boundary of P. We regard all points in Ed as row vectors and use qT for the column vector that is the transpose of the row vector q. Moreover, Œq1 ; : : : ; qd is the (square) matrix with the i th row qi . Choosing a.d 1/-dimensional simplex of CP to be positively oriented, one can check whether the orientation of an arbitrary .d 1/-dimensional simplex convfcF0 ; cF1 ; : : : ; cFd 1 g of CP (generated by the given sequence of its vertices), is positive or negative. Let sign convfcF0 ; cF1 ; : : : ; cFd 1 g be equal to 1 (resp., 1) if the orientation of the .d 1/-dimensional simplex convfcF0 ; cF1 ; : : : ; cFd 1 g is positive (resp., negative).

2.6 Proof of Theorem 1.5.3

51

Definition 2.6.2. The signed volume V .P/ of P is defined as 1 dŠ F

X

sign convfcF0 ; cF1 ; : : : ; cFd 1 g detŒcF0 ; cF1 ; : : : ; cFd 1 ; (2.40)

0 F1 Fd 1

where the sum is taken over all flags of faces F0 F1 Fd 1 of P, and detŒ is the determinant function. The following is clear. Lemma 2.6.3. V .P/ D

X 1 d Š F F 0

sign convfcF0 ; cF1 ; : : : ; cFd 1 g cF0 ^ cF1 ^ ^ cFd 1 ; d 1

where ^ stands for the wedge product of vectors. Moreover, one can choose the orientation of the boundary of P such that V .P/ D vold .P/, where vold ./ refers to the d -dimensional volume measure in Ed ; d 2.

2.6.2 The Volume Force of Convex Polytopes We wish to compute the gradient of V .P/, where P D convfp1 ; p2 ; : : : ; pn g is regarded as a function of its vertices p1 ; p2 ; : : : ; pn . To achieve this we consider an arbitrary path p.t/ D p C tp0 in the space of the configurations p WD .p1 ; p2 ; : : : ; pn /, where p0 WD .p01 ; p02 ; : : : ; p0n / (Fig. 2.8). Based on Definitions 2.6.1, 2.6.2, and Lemma 2.6.3 we introduce V .P.t// as a function of t (with t being an arbitrary real with sufficiently small absolute value) via 1 dŠ F D

X

sign convfcF0 .t/; : : : ; cFd 1 .t/g detŒcF0 .t/; : : : ; cFd 1 .t/

0 F1 Fd 1

1 dŠ F

X

sign convfcF0 .t/; : : : ; cFd 1 .t/g cF0 .t/ ^ ^ cFd 1 .t/;

0 F1 Fd 1

P where cF .t/ WD k1 kj D1 pij .t/ for any face F D convfpi1 ; : : : ; pik g of P. Clearly, d V .P.t// of V .P.t// at V .P.0// D V .P/. Moreover, evaluating the derivative dt t D 0, collecting terms, and using the anticommutativity of the wedge product we get that 1 X d V .P.t// jt D0 D Ni ^ p0i ; dt d Š i D1 n

(2.41)

52

2 Proofs on Unit Sphere Packings

p`i

pi + p`i

p`j

pj + p`j

Fig. 2.8 Deformation of the convex polytope P under the motion given by p0

where each Ni is some linear combination of wedge products of d 1 vectors pj with pj and pi sharing a common face. Definition 2.6.4. We call N WD .N1 ; N2 ; : : : ; Nn / the volume force of the d -dimensional convex polytope P Ed with n vertices. The following are some simple properties of the volume force. We leave the rather straightforward proofs to the reader. Lemma 2.6.5. Let N WD .N1 ; N2 ; : : : ; Nn / be the volume force of the d -dimensional convex polytope P Ed ; d 2 with vertices p1 ; p2 ; : : : ; pn . Then the following hold. (1) Each Ni is only a function of the vertices that share a face with pi , but not pi itself. (2) Assume that the origin o of Ed is the barycenter of P; moreover, let T W Ed ! Ed be an orthogonal linear map satisfying T .P/ D P. If T .pi / D pj , then T .Ni / D Nj . For more details and examples on volume forces we refer the interested reader to the proper sections in [56].

2.6.3 Critical Volume Condition Let P WD convfp1 ; p2 ; : : : ; pn g be a d -dimensional convex polytope in Ed ; d 2 with vertices p WD .p1 ; p2 ; : : : ; pn /. Let G be a graph defined on this vertex set p. Here, G may or may not consist of the edges of P. We think of the edges of G as defining those pairs of vertices of P that are constrained not to get closer. In the terminology of the geometry of rigid tensegrity frameworks each edge of G is a strut. (For more information on rigid tensegrity frameworks and the basic terminology used there we refer the interested reader to [162].) Let p0 WD .p01 ; p02 ; : : : ; p0n / be an infinitesimal flex of G.p/, where G.p/ refers to the realization of G over the point configuration p. That is, for each edge (strut) fi; j g of G we have .pi pj / .p0i p0j / 0;

(2.42)

where “” denotes the standard inner product (also called the “dot product”) in Ed .

2.6 Proof of Theorem 1.5.3

53

Let e denote the number of edges of G. Then the rigidity matrix R.p/ of G.p/ is the e nd matrix whose row corresponding to the edge fi; j g of G consists of the coordinates of d -dimensional vectors within a sequence of n vectors such that all the coordinates are zero except maybe the ones that correspond to the coordinates of the vectors pi pj and pj pi listed on the i th and j th position. Another way to introduce R.p/ is the following. Let f W End ! Ee be the map defined by x D .x1 ; x2 ; : : : ; xn / ! .: : : ; kxi xj k2 ; : : : /. Then it is immediate that 1 d 2 d x f jxDp D R.p/. Now, we can rewrite the inequalities of (2.42) in terms of the rigidity matrix R.p/ of G.p/ (using the usual matrix multiplication applied to R.p/ and the indicated column vector) as follows, R.p/.p0 /T 0;

(2.43)

where the inequality is meant for each coordinate. For each edge fi; j g of G, let !ij be a scalar. We collect all such scalars into a single row vector called the stress ! WD .: : : ; !ij ; : : : / corresponding to the rows of the matrix R.p/. Append the volume force N WD .N1 ; N2 ; : : : ; Nn / as the last row O onto R.p/ to get a new matrix R.p/, which we call the augmented rigidity matrix. 0 T O So, when performing the matrix multiplication R.p/.p / , we find that the result is a column vector of length e C 1 having .pi pj / .p0i p0j / on the position P corresponding to the edge fi; j g of G, and having nkD1 Nk p0k on the .e C 1/st position. Also, it is easy to see that 0 O .!; 1/R.p/ D @: : : ;

X

1 !ij .pi pj / C Ni ; : : : A ;

(2.44)

j

where each sum is taken over all pj adjacent to pi in G, and we collect d coordinates at a time. Definition 2.6.6. Let N D .N1 ; N2 ; : : : ; Nn / be the volume force of the d dimensional convex polytope P Ed ; d 2 with vertices p D .p1 ; p2 ; : : : ; pn /. We say that the stress ! D .: : : ; !ij ; : : : / resolves N if for each i we have that P O j !ij .pi pj / C Ni D o or, equivalently, .!; 1/R.p/ D o, where o denotes the zero vector. Definition 2.6.7. The d -dimensional convex polytope P Ed ; d 2 and the graph G defined on the vertices of P satisfy the critical volume condition if the volume force N can be resolved by a stress ! D .: : : ; !ij ; : : : / such that for each edge fi; j g of G, !ij < 0. Theorem 2.6.8. Let the d -dimensional convex polytope P Ed ; d 2 and the strut graph G, defined on the vertices of P, satisfy the critical volume condition. Moreover, let p0 D .p01 ; p02 ; : : : ; p0n / be an infinitesimal flex of the strut framework G.p/ (i.e., let p0 satisfy (2.42)). Then

54

2 Proofs on Unit Sphere Packings

1 X d Ni ^ p0i 0 V .P.t// jt D0 D dt d Š i D1 n

with equality if and only if .pi pj / .p0i p0j / D 0 for each edge fi; j g of G. Proof. The assumptions, (2.44), the associativity of matrix multiplication, and (2.41) imply in a straightforward way that 0 T O 0 D o p0 D .!; 1/R.p/.p / D

X

!ij .pi pj / .p0i p0j / C

X fi;j g

!ij .pi pj / .p0i p0j / C

n X i D1

Ni ^ p0i

Ni p0i

i D1

fi;j g

D

n X

n X

Ni ^ p0i D

i D1

d V .P.t// jt D0 ; dt

where Ni is regarded as a d -dimensional vector so that Ni ^ p0i can be interpreted as the standard inner product Ni pi , with appropriate identification of bases. We clearly get equality if and only if .pi pj / .p0i p0j / D 0 for each edge fi; j g of G.

2.6.4 Strictly Locally Volume Expanding Convex Polytopes The following definition recalls standard terminology from the theory of rigid tensegrity frameworks. (See [78] for more information.) Consider now just the bar graph G, which is the graph G with all the struts changed to bars, and take its realization G.p/ sitting over the point configuration p D .p1 ; p2 ; : : : ; pn /. (Here bars mean edges whose lengths are constrained not to change.) We say that the infinitesimal motion p0 D .p01 ; p02 ; : : : ; p0n / is an infinitesimal flex of G.p/ if for each edge (bar) fi; j g of G, we have .pi pj / .p0i p0j / D 0: This is the same as saying R.p/.p0 /T D o for the rigidity matrix R.p/. Definition 2.6.9. We say that p0 is a trivial infinitesimal flex if p0 is a (directional) derivative of an isometric motion of Ed ; d 2. We say that G.p/ (resp., G.p/) is infinitesimally rigid if G.p/ (resp., G.p/) has only trivial infinitesimal flexes. Theorem 2.6.10. Let the d -dimensional convex polytope P Ed ; d 2 and the strut graph G, defined on the vertices of P, satisfy the critical volume condition and assume that the bar framework G.p/ is infinitesimally rigid. Then

2.6 Proof of Theorem 1.5.3

55

1 X d Ni ^ p0i > 0 V .P.t// jt D0 D dt d Š i D1 n

for every non-trivial infinitesimal flex p0 D .p01 ; p02 ; : : : ; p0n / of the strut framework G.p/. P d Proof. By Theorem 2.6.8 we have that dt V .P.t// jt D0 D d1Š niD1 Ni ^ p0i 0. d V .P.t// jt D0 D 0, then applying Theorem 2.6.8 again, p0 D .p01 ; p02 ; : : : ; If dt 0 pn / must be an infinitesimal flex of the bar framework G.p/. However, then by the infinitesimal rigidity of G.p/, this would imply that p0 is trivial. Thus, d V .P.t// jt D0 > 0. dt The following definition leads us to the core part of this section. Definition 2.6.11. Let P Ed ; d 2 be a d -dimensional convex polytope and let G be a strut graph defined on the vertices p D .p1 ; p2 ; : : : ; pn / of P. We say that P is strictly locally volume expanding over G, if there is an " > 0 with the following property. For every q D .q1 ; q2 ; : : : ; qn / satisfying kpi qi k < " for all i D 1; : : : ; n

(2.45)

kpi pj k kqi qj k for each edge fi; j g of G;

(2.46)

and

we have V .P/ V .Q/ (where V .Q/ is defined via (2.39) and (2.40) substituting q for p) with equality only when P is congruent to Q, where Q is the polytope generated by the simplices of the barycenters in (2.39) using q instead of p. Theorem 2.6.12. Let the d -dimensional convex polytope P Ed ; d 2 and the strut graph G, defined on the vertices of P, satisfy the critical volume condition and assume that the bar framework G.p/ is infinitesimally rigid. Then P is strictly locally volume expanding over G. Proof. The inequalities (2.46) define a semialgebraic set X in the space of all configurations f.q1 ; q2 ; : : : ; qn /jqi 2 Ed ; i D 1; : : : ; ng. Suppose there is no " as in the conclusion. Add V .P/ V .Q/ to the constraints defining X . By Wallace [179] (see [78]) there is an analytic path p.t/ D .p1 .t/; p2 .t/; : : : ; pn .t//; 0 t < 1, with p.0/ D p and p.t/ 2 X , p.t/ not congruent to p.0/ for 0 < t < 1. So, kpi pj k kpi .t/ pj .t/k for each edge fi; j g of G and V .P/ V .P.t// for 0 t < 1:

(2.47) (2.48)

56

2 Proofs on Unit Sphere Packings

Then after suitably adjusting p.t/ by congruences (as in [78] as well as [80]) we can define p0 WD

d k p.t/ jt D0 dt k

for the smallest k that makes p0 a non-trivial infinitesimal flex. (Such k exists by the argument in [78] as well as [80]). Because (2.47) holds we see that p0 is a non-trivial infinitesimal flex of G.p/ and (2.48) implies that d V .P.t// jt D0 0: dt But this contradicts Theorem 2.6.10, finishing the proof of Theorem 2.6.12.

2.6.5 From Critical Volume Condition to Uniform Stability Here we start with the assumptions of Theorem 1.5.3 and apply Theorem 2.6.12 to each Pi and GP restricted to the vertices of Pi , 1 i m. Then let "0 > 0 be the smallest " > 0 guaranteed by the strict locally volume expanding property of Theorem 2.6.12. All but a finite number of tiles are fixed. The tiles that are free to move are confined to a region of fixed volume in Ed ; d 2. Each Pi is strictly locally volume expanding, therefore the volume of each of the tiles must be fixed. But the strict condition implies that the motion of each tile must be an isometry. Because the tiling is face-to-face and the vertices are given by GP we conclude inductively (on the number of tiles) that each vertex of GP must be fixed. Thus, P is uniformly stable with respect to "0 introduced above, finishing the proof of Theorem 1.5.3.

Chapter 3

Contractions of Sphere Arrangements

Abstract Extending the tradition of studying packings of spheres we investigate the monotonicity of volume under contractions of arbitrary arrangements of spheres. The research on this fundamental topic started with the conjecture of E. T. Poulsen and M. Kneser in the late 1950s. In this chapter we survey the status of the long-standing Kneser–Poulsen conjecture in Euclidean as well as in non-Euclidean spaces.

3.1 The Kneser–Poulsen Conjecture Recall that kk denotes the standard Euclidean norm of the d -dimensional Euclidean space Ed . So, if pi ; pj are two points in Ed , then kpi pj k denotes the Euclidean distance between them. It is convenient to denote the (finite) point configuration consisting of the points p1 ; p2 ; : : : ; pN in Ed by p D .p1 ; p2 ; : : : ; pN /. Now, if p D .p1 ; p2 ; : : : ; pN / and q D .q1 ; q2 ; : : : ; qN / are two configurations of N points in Ed such that for all 1 i < j N the inequality kqi qj k kpi pj k holds, then we say that q is a contraction of p. If q is a contraction of p, then there may or may not be a continuous motion p.t/ D .p1 .t/; p2 .t/; : : : ; pN .t//, with pi .t/ 2 Ed for all 0 t 1 and 1 i N such that p.0/ D p and p.1/ D q, and kpi .t/ pj .t/k is monotone decreasing for all 1 i < j N . When there is such a motion, we say that q is a continuous contraction of p. Finally, let Bd Œpi ; ri denote the (closed) d -dimensional ball centered at pi with radius ri in Ed and let vold ./ represent the d -dimensional volume (Lebesgue measure) in Ed . In 1954 Poulsen [158] and in 1955 Kneser [140] independently conjectured the following for the case when r1 D D rN (Fig. 3.1).

K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 3, © Springer International Publishing Switzerland 2013

57

58

3 Contractions of Sphere Arrangements

p1

p3

q3 q1 Contractive

p2

Mapping

q2 q4

p4

Fig. 3.1 The Kneser-Poulsen conjecture in E2

Conjecture 3.1.1. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed , then ! ! N N [ [ d d vold B Œpi ; ri vold B Œqi ; ri : i D1

i D1

A similar conjecture was proposed by Klee and Wagon [138] in 1991. Conjecture 3.1.2. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed , then ! ! N N \ \ d d B Œpi ; ri vold B Œqi ; ri : vold i D1

i D1

Actually, Kneser seems to be the one who has generated a great deal of interest in the above conjectures also via private letters written to a number of mathematicians. For more details on this see, for example, [138].

3.2 The Kneser–Poulsen Conjecture for Continuous Contractions For a given point configuration p D .p1 ; p2 ; : : : ; pN / in Ed and radii r1 ; r2 ; : : : ; rN consider the following sets, Vi D fx 2 Ed j for all j; kx pi k2 ri2 kx pj k2 rj2 g; Vi D fx 2 Ed j for all j; kx pi k2 ri2 kx pj k2 rj2 g:

3.2 The Kneser–Poulsen Conjecture for Continuous Contractions

59

Fig. 3.2 The nearest (resp., farthest) point truncated Voronoi cell decomposition of the union (resp., intersection) of four disks in E2

The set Vi (resp., Vi ) is called the nearest (resp., farthest) point Voronoi cell of the point pi . (For a detailed discussion on nearest as well as farthest point Voronoi cells we refer the interested reader to [96] and [171].) We now restrict each of these sets as follows. Vi .ri / D Vi \ Bd Œpi ; ri ; Vi .ri / D Vi \ Bd Œpi ; ri : We call the set Vi .ri / (resp., Vi .ri /) the nearest (resp., farthest) point truncated Voronoi cell of the point pi (Fig. 3.2). For each i ¤ j let Wij D Vi \ Vj and W ij D Vi \ Vj . The sets Wij and W ij are the walls between the nearest and farthest point Voronoi cells. Finally, it is natural to define the relevant truncated walls as follows. Wij .pi ; ri / D Wij \ Bd Œpi ; ri D Wij .pj ; rj / D Wij \ Bd Œpj ; rj ;

W ij .pi ; ri / D W ij \ Bd Œpi ; ri D W ij .pj ; rj / D W ij \ Bd Œpj ; rj : The following formula discovered by Csik´os [84] proves Conjecture 3.1.1 as well as Conjecture 3.1.2 for continuous contractions in a straighforward way in any dimension. (Actually, the planar case of the Kneser–Poulsen conjecture under continuous contractions has been proved independently in [62, 71, 83], and [22].)

60

3 Contractions of Sphere Arrangements

Theorem 3.2.1. Let d 2 and let p.t/; 0 t 1 be a smooth motion of a point configuration in Ed such that for each t, the points of the configuration are pairwise distinct. Then ! N [ d d vold B Œpi .t/; ri dt i D1 X d D dij .t/ vold 1 Wij .pi .t/; ri / ; dt 1i <j N ! N \ d d vold B Œpi .t/; ri dt i D1 X d dij .t/ vold 1 W ij .pi .t/; ri / ; D dt 1i <j N where dij .t/ D kpi .t/ pi .t/k. On the one hand, Csik´os [85] managed to generalize his formula to configurations of balls called flowers which are sets obtained from balls with the help of operations \ and [. This work extends to hyperbolic as well as spherical space. On the other hand, Csik´os [86] has succeeded in proving a Schl¨afli-type formula for polytopes with curved faces lying in pseudo-Riemannian Einstein manifolds, which can be used to provide another proof of Conjecture 3.1.1 as well as Conjecture 3.1.2 for continuous contractions (for more details see [86]).

3.3 On the Kneser–Poulsen Conjecture for General Contractions In the recent paper [48] the author and Connelly proved Conjecture 3.1.1 as well as Conjecture 3.1.2 in the Euclidean plane. Thus, we have the following theorem. Theorem 3.3.1. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in E2 , then ! ! N N [ [ vol2 B2 Œpi ; ri vol2 B2 Œqi ; ri I i D1

i D1

moreover, vol2

N \ i D1

! B2 Œpi ; ri vol2

N \ i D1

! B2 Œqi ; ri :

3.3 On the Kneser–Poulsen Conjecture for General Contractions

61

In fact, the paper [48] contains a proof of an extension of the above theorem to flowers as well. In what follows we give an outline of the three-step proof published in [48] by phrasing it through a sequence of theorems each being higherdimensional. Voronoi cells play an essential role in our proofs of Theorems 3.3.2 and 3.3.3. Theorem 3.3.2. Consider N moving closed d -dimensional balls Bd Œpi .t/; ri with 1 i N; 0 t 1 in Ed ; dS 2. If Fi .t/ is the contribution of the d i th ball to the boundary of the union N i D1 B Œpi .t/; ri (resp., of the intersection TN d i D1 B Œpi .t/; ri ), then X 1 svold 1 .Fi .t// r 1i N i decreases (resp., increases) in t under any analytic contraction p.t/ of the center points, where 0 t 1 and svold 1 .: : : / refers to the relevant .d 1/-dimensional surface volume. Theorem 3.3.3. Let the centers of the closed d -dimensional balls Bd Œpi ; ri , 1 i N lie in the .d 2/-dimensional affine subspace L of Ed ;S d 3. If Fi stands for d the contribution of the i th ball to the boundary of the union N i D1 B Œpi ; ri (resp., TN d of the intersection i D1 B Œpi ; ri ), then

vold 2

!

N [

d 2

B

Œpi ; ri D

N \

d 2

i D1

resp:; vold 2

i D1

!

B

1 X 1 svold 1 .Fi / 2 1i N ri

! 1 X 1 Œpi ; ri D svold 1 .Fi / ; 2 1i N ri

where Bd 2 Œpi ; ri D Bd Œpi ; ri \ L; 1 i N . Theorem 3.3.4. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed ; d 1, then there is an analytic contraction p.t/ D .p1 .t/; : : : ; pN .t//; 0 t 1 in E2d such that p.0/ D p and p.1/ D q. Note that Theorems 3.3.2–3.3.4 imply Theorem 3.3.1 in a straighforward way. Also, we note that Theorem 3.3.4 (called the Leapfrog Lemma) cannot be improved; namely, it has been shown in [20] that there exist point configurations q and p in Ed , actually constructed in the way suggested in [48], such that q is a contraction of p in Ed and there is no continuous contraction from p to q in E2d 1 . In order to describe a more complete picture of the status of the Kneser–Poulsen conjecture we mention two additional corollaries obtained from the proof published in [48] and just outlined above. (For more details see [48].)

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3 Contractions of Sphere Arrangements

Theorem 3.3.5. Let p D .p1 ; p2 ; : : : ; pN / and q D .q1 ; q2 ; : : : ; qN / be two point configurations in Ed such that q is a piecewise-analytic contraction of p in Ed C2 . Then the conclusions of Conjecture 3.1.1 as well as Conjecture 3.1.2 hold in Ed . The following generalizes a result of Gromov in [108], who proved it in the case N d C 1. Theorem 3.3.6. If q D .q1 ; q2 ; : : : ; qN / is an arbitrary contraction of p D .p1 ; p2 ; : : : ; pN / in Ed and N d C 3, then both Conjectures 3.1.1 and 3.1.2 hold. As a next step it would be natural to investigate the case N D d C 4.

3.4 Kneser–Poulsen-Type Theorems in Non-Euclidean Spaces It is somewhat surprising that in spherical space for the specific radius of balls (i.e., spherical caps) one can find a proof of both Conjectures 3.1.1 and 3.1.2 in all dimensions. The magic radius is 2 and the following theorem describes the desired result in details. Theorem 3.4.1. If a finite set of closed d -dimensional balls of radius 2 (i.e., of closed hemispheres) in the d -dimensional spherical space Sd ; d 2 is rearranged so that the (spherical) distance between each pair of centers does not increase, then the (spherical) d -dimensional volume of the intersection does not decrease and the (spherical) d -dimensional volume of the union does not increase. The method of the proof published by the author and Connelly in [49] can be described as follows. First, one can use a leapfrog lemma to move one configuration to the other in an analytic and monotone way, but only in higher dimensions. Then the higher-dimensional balls have their combined volume (their intersections or unions) change monotonically, a fact that one can prove using Schl¨afli’s differential formula. Then one can apply an integral formula to relate the volume of the higherdimensional object to the volume of the lower-dimensional object, obtaining the volume inequality for the more general discrete motions. The following statement is a corollary of Theorem 3.4.1, the Euclidean part of which has been proved independently by Alexander [4], Capoyleas and Pach [72], and Sudakov [173]. For the sake of completeness in what follows, we recall the notion of spherical mean width, which is most likely less known than its widely used Euclidean counterpart. Let Sd be the d -dimensional unit sphere centered at the origin in Ed C1 . A spherically convex body is a closed, spherically convex subset of Sd with interior points and lying in some closed hemisphere, thus, the intersection of Sd with a .d C 1/-dimensional closed convex cone of Ed C1 different from Ed C1 . Recall that Svold .: : : / denotes the spherical Lebesgue measure on Sd , and recall that .d C 1/!d C1 D Svold .Sd /. Moreover, as usual we denote the standard inner

3.4 Kneser–Poulsen-Type Theorems in Non-Euclidean Spaces

63

product of Ed C1 by h; i, and for u 2 Sd we write u? WD fx 2 Ed C1 W hu; xi D 0g for the orthogonal complement of linfug. For a spherically convex body K, the polar body is defined by K WD fu 2 Sd W hu; vi 0 for all v 2 Kg: It is also spherically convex, but need not have interior points. The number U.K/ WD

1 Svold .fu 2 Sd W u? \ K 6D ;g/ 2

can be considered as the spherical mean width of K. Obviously, a vector u 2 Sd satisfies u 2 K [ .K / if and only if u? does not meet the interior of K, hence .d C 1/!d C1 2Svold .K / D 2U.K/:

(3.1)

Now, (3.1) and Theorem 3.4.1 imply the following theorem in a rather straightforward way. Theorem 3.4.2. Let p D .p1 ; p2 ; : : : ; pN / be N points on a closed hemisphere of Sd ; d 2 (resp., points in Ed ; d 2), and let q D .q1 ; q2 ; : : : ; qN / be a contraction of p in Sd (resp., in Ed ). Then the spherical mean width (resp., mean width) of the spherical convex hull (resp., convex hull) of q is less than or equal to the spherical mean width (resp., mean width) of the spherical convex hull (resp., convex hull) of p. We note that Theorem 3.4.1 extends to flowers as well; moreover, a positive answer to the following problem would imply that both Conjectures 3.1.1 and 3.1.2 hold for circles in S2 (for more details on this see [49]). Problem 3.4.3. Suppose that p D .p1 ; p2 ; : : : ; pN / and q D .q1 ; q2 ; : : : ; qN / are two point configurations in S2 . Then prove or disprove that there is a monotone piecewise-analytic motion from p D .p1 ; p2 ; : : : ; pN / to q D .q1 ; q2 ; : : : ; qN / in S4 . Note that in fact, Theorem 3.4.1 states a volume inequality between two spherically convex polytopes satisfying some metric conditions. The following problem searches for a natural analogue of that in the hyperbolic 3-space H3 . In order to state it properly we recall the following. Let A and B be two planes in H3 and let AC (resp., B C ) denote one of the two closed halfspaces bounded by A (resp., B) such that the set AC \ B C is nonempty. Recall that either A and B intersect or A is parallel to B or A and B have a line perpendicular to both of them. Now, the dihedral angle AC \B C means not only the set in question, but also refers to the standard angular measure of the corresponding angle between A and B in the first case, it refers to 0 in the second case, and finally, in the third case it refers to the negative of the hyperbolic distance between A and B.

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3 Contractions of Sphere Arrangements

Problem 3.4.4. Let P and Q be compact convex polyhedra of H3 with P (resp., Q) C C C being the intersection of the closed halfspaces HP;1 ; HP;2 ; : : : ; HP;N (resp., C C C C C HQ;1 ; HQ;2 ; : : : ; HQ;N ). Assume that the dihedral angle HQ;i \HQ;j (containing Q) C C \ HP;j (containing P) is at least as large as the corresponding dihedral angle HP;i for all 1 i < j N . Then prove or disprove that the volume of P is at least as large as the volume of Q. Using Andreev’s version [8, 9] of the Koebe–Andreev–Thurston theorem and Schl¨afli’s differential formula the author [35] proved the following partial analogue of Theorem 3.4.1 in H3 . Theorem 3.4.5. Let P and Q be nonobtuse-angled compact convex polyhedra of the same simple combinatorial type in H3 . If each inner dihedral angle of Q is at least as large as the corresponding inner dihedral angle of P, then the volume of P is at least as large as the volume of Q.

3.5 The Kneser–Poulsen Conjecture for Large Equal Radii Let K be a convex body in Ed . Recall that the mean width Md .K/ of K (up to multiplication by a dimensional constant) can be defined by Z Md .K/ WD maxfhu; xi j x 2 Kgd u; Sd 1

where d u denotes the spherical volume element on Sd 1 generated by the .d 1/dimensional spherical Lebesgue measure on Sd 1 with the total measure of Sd 1 being equal to d!d . The first formula of the following theorem is due to Capoyleas and Pach [72] and the second one was conjectured by Csik´os and proved by Gorbovickis [107]. Theorem 3.5.1. Let p D .p1 ; p2 ; : : : ; pN / be a point configuration in Ed ; d 2. Then the following asymptotic formulas hold (for r ! C1): ! N [ d vold B Œpi ; r D !d r d C Md .convfp1 ; p2 ; : : : ; pN g/ r d 1 C o.r d 1 / i D1

and vold

N \

! B Œpi ; r D !d r d Md .convfp1 ; p2 ; : : : ; pN g/ r d 1 C o.r d 1 /: d

i D1

Also, Gorbovickis [107] has proved the following strengthening (i.e., strict version) of Theorem 3.4.2 in the Euclidean case. Here it is natural to conjecture that the spherical version of that theorem is true as well.

3.6 Alexander’s Conjecture Revisited

65

Theorem 3.5.2. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed ; d 2, then Md .convfq1 ; q2 ; : : : ; qN g/ Md .convfp1 ; p2 ; : : : ; pN g/ and if the point configurations q and p are not congruent, then the inequality is strict. Thus, Theorems 3.5.1 and 3.5.2 imply in a straigthforward way that the Kneser– Poulsen Conjecture holds for sufficiently large equal radii (for more details see [107]). Theorem 3.5.3. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed ; d 2, then there exists r0 > 0 such that for any r r0 , vold

N [

!

N [

B Œpi ; r vold d

i D1

! d

B Œqi ; r

i D1

and vold

N \ i D1

! B Œpi ; r vold d

N \

! d

B Œqi ; r ;

i D1

and if the point configurations q and p are not congruent, then the inequality is strict.

3.6 Alexander’s Conjecture Revisited It seems that in the Euclidean plane, for the case of the intersection of congruent disks, one can sharpen the results proved by the author and Connelly [48]. Namely, Alexander [4] conjectures the following. Conjecture 3.6.1. Under arbitrary contraction of the center points of finitely many congruent disks in the Euclidean plane, the perimeter of the intersection of the disks cannot decrease. The analogous question for the union of congruent disks has a negative answer, as was observed by Habicht and Kneser long ago (for details see [48]). In [58] some supporting evidence for the above conjecture of Alexander has been collected; in particular, the following theorem was proved. Theorem 3.6.2. Alexander’s conjecture holds for continuous contractions of the center points and it holds up to 4 congruent disks under arbitrary contractions of the center points.

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3 Contractions of Sphere Arrangements

We note that Alexander’s conjecture does not hold for incongruent disks (even under continuous contractions of their center points) as shown in [58]. Finally we remark that if Alexander’s conjecture were true, then it would be a rare instance of an asymmetry between intersections and unions for Kneser–Poulsen-type questions.

Chapter 4

Proofs on Contractions of Sphere Arrangements

Abstract First, we give a proof of the long-standing Kneser–Poulsen conjecture in the Euclidean plane. Although that result is 2-dimensional its proof is higher dimensional. Second, we prove an analogue of the Kneser–Poulsen conjecture for hemispheres in spherical d -space. Third, we give a proof of a Kneser–Poulsen-type theorem for convex polyhedra in hyperbolic 3-space.

4.1 Proof of Theorem 3.3.2: Weighted Surface Volume Theorem 3.3.2. Consider N moving closed d -dimensional balls Bd Œpi .t/; ri with 1 i N; 0 t 1 in Ed ; dS 2. If Fi .t/ is the contribution of the d i th ball to the boundary of the union N i D1 B Œpi .t/; ri (resp., of the intersection TN d i D1 B Œpi .t/; ri ), then X 1 svold 1 .Fi .t// r 1i N i decreases (resp., increases) in t under any analytic contraction p.t/ of the center points, where 0 t 1 and svold 1 .: : : / refers to the relevant .d 1/-dimensional surface volume. As a first step in our proof of Theorem 3.3.2, we recall the following underlying system of (truncated) Voronoi cells. For a given point configuration p D .p1 ; p2 ; : : : ; pN / in Ed and radii r1 ; r2 ; : : : ; rN consider the following sets, Vi D fx 2 Ed j for all j; kx pi k2 ri2 kx pj k2 rj2 g; Vi D fx 2 Ed j for all j; kx pi k2 ri2 kx pj k2 rj2 g:

K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 4, © Springer International Publishing Switzerland 2013

67

68

4 Proofs on Contractions of Sphere Arrangements

The set Vi (resp., Vi ) is called the nearest (resp., farthest) point Voronoi cell of the point pi . We now restrict each of these sets as follows, Vi .ri / D Vi \ Bd Œpi ; ri ; Vi .ri / D Vi \ Bd Œpi ; ri : We call the set Vi .ri / (resp., Vi .ri /) the nearest (resp., farthest) point truncated Voronoi cell of the point pi . For each i ¤ j let Wij D Vi \ Vj and W ij D Vi \ Vj . The sets Wij and W ij are the walls between the nearest point and farthest point Voronoi cells. Finally, it is natural to define the relevant truncated walls as follows. Wij .pi ; ri / D Wij \ Bd Œpi ; ri D Wij .pj ; rj / D Wij \ Bd Œpj ; rj ; W ij .pi ; ri / D W ij \ Bd Œpi ; ri D W ij .pj ; rj / D W ij \ Bd Œpj ; rj : Second, for each i D 1; 2; : : : ; N and 0 s, define ri .s/ D

q

1 d ri .s/ D : ds 2ri .s/ Now, define r.s/ D .r1 .s/; : : : ; rN .s//, and introduce Vd .t; s/ WD vold BdS Œp.t/; r.s/ ; and

V d .t; s/ WD vold BdT Œp.t/; r.s/

as functions of the variables t and s, where BdS Œp.t/; r.s/ WD

N [

Bd Œpi .t/; ri .s/;

i D1

and BdT Œp.t/; r.s/ WD

N \ i D1

Throughout we assume that all ri > 0.

Bd Œpi .t/; ri .s/:

ri2 C s. Clearly, (4.1)

4.1 Proof of Theorem 3.3.2: Weighted Surface Volume

69

Lemma 4.1.1. Let d 2 and let p.t/; 0 t 1 be a smooth motion of a point configuration in Ed such that for each t, the points of the configuration are pairwise distinct. Then the volume functions Vd .t; s/ and V d .t; s/ are continuously differentiable in t and s simultaneously, and for any fixed t, the nearest point and farthest point Voronoi cells are constant. Proof. Let t D t0 be fixed. Then recall that the point x belongs to the Voronoi cell Vi .t0 ; s/ (resp., Vi .t0 ; s/), when for all j , kx pi .t0 /k2 kx pj .t0 /k2 ri .s/2 C rj .s/2 is non-positive (resp., non-negative). But ri .s/2 rj .s/2 D ri2 rj2 is constant. So each Vi .t0 ; s/ and Vi .t0 ; s/ is a constant function of s. As p.t/ is continuously differentiable, therefore the partial derivatives of Vd .t; s/ and V d .t; s/ with respect to t exist and are continuous by Theorem 3.2.1. Each ball Bd Œpi .t/; ri .s/; d 2 is strictly convex. Hence, the .d 1/-dimensional surface volume of the boundaries of BdS Œp.t/; r.s/ and BdT Œp.t/; r.s/ are continuous functions of s, and the partial derivatives of Vd .t; s/ and V d .t; s/ with respect to s exist and are continuous. Thus, Vd .t; s/ and V d .t; s/ are both continuously differentiable with respect to t and s simultaneously. Lemma 4.1.2. Let p.t/; 0 t 1 be an analytic motion of a point configuration in Ed ; d 2. Then there exists an open dense set U in Œ0; 1 .0; 1/ such that for any .t; s/ 2 U the following hold. X d

@ @2 Vd .t; s/ D dij .t/ vold 1 Wij .pi .t/; ri .s// ; @t@s dt @s 1i <j N and X

@2 d d @ V .t; s/ D dij .t/ vold 1 W ij .pi .t/; ri .s// : @t@s dt @s 1i <j N Hence, if p.t/ is contracting, then @s@ Vd .t; s/ is monotone decreasing in t, and @ V d .t; s/ is monotone increasing in t. @s Proof. Given that p.t/; 0 t 1 is an analytic function of t, we wish to define an open dense set U in Œ0; 1.0; 1/, where the volume functions Vd .t; s/ and V d .t; s/ are analytic in t and s simultaneously. Lemma 4.1.1 implies that the Voronoi cells Vi and Vi are functions of t alone. Moreover, clearly there are only a finite number of values of t in the interval Œ0; 1, where the combinatorial type of the above Voronoi cells changes. The volume of the truncated Voronoi cells Vi .ri .s// and Vi .ri .s// are obtained from the volume of the d -dimensional Euclidean ball of radius ri .s/ by removing or adding the volumes of the regions obtained by conning over the walls Wij .pi .t/; ri .s// or W ij .pi .t/; ri .s// from the point pi .t/. By induction on d , starting at d D 1, each Wij and W ij is an analytic function of t and s, when the ball of radius ri .s/ is not tangent to any of the faces of Vi or Vi . So, for any fixed t the ball of radius ri .s/ will not be tangent to any of the faces Vi or Vi for

70

4 Proofs on Contractions of Sphere Arrangements

all but a finite number of values of s. Thus, we define U to be the set of those .t; s/, where for some open interval about t in Œ0; 1, the combinatorial type of the Voronoi cells is constant and for all i , the ball of radius ri .s/ is not tangent to any of the faces of Vi or Vi . We also assume that the points of the configuration p.t/ are distinct for any .t; s/ 2 U . If, for i ¤ j and for infinitely many values of t in the interval Œ0; 1, pi .t/ D pj .t/, then they are the same point for all t, and those points may be identified. Then the set U is open and dense in Œ0; 1 .0; 1/ and the volume functions Vd .t; s/ and V d .t; s/ are analytic in t and s. Thus, the formulas for the mixed partial derivatives in Lemma 4.1.2 follow from the definition of U and from Theorem 3.2.1. (Note also that here we could interchange the order of partial differentiation with respect to the variables t and s.) To show that @s@ Vd .t; s/ and @s@ V d .t; s/ are monotone, suppose they are not. We show a contradiction. If we perturb s slightly to s0 say, then using the formulas for the mixed partial derivatives in Lemma 4.1.2 we get that the partial derivative of @s@ Vd .t; s/ and @s@ V d .t; s/ with respect to t exists and has the appropriate sign, except for a finite number of values of t for s D s0 . (Here we have also used the following rather obvious monotonicity property of the walls: Wij .pi .t/; ri .s// Wij .pi .t/; ri .s // and W ij .pi .t/; ri .s// W ij .pi .t/; ri .s // for any s s .) Since @s@ Vd .t; s/ and @s@ V d .t; s/ are continuous as a function of t at s D s0 by Lemma 4.1.1, they are monotone. But the functions at s0 approximate the functions at s (again by Lemma 4.1.1) providing the contradiction. So, @s@ Vd .t; s/ and @s@ V d .t; s/ are indeed monotone. This completes the proof of Lemma 4.1.2. First, note that Fi .t/ D Vi .t; 0/ \ bd BdS Œp.t/; r.0/

(4.2)

resp:; Fi .t/ D Vi .t; 0/ \ bd BdT Œp.t/; r.0/ :

(4.3)

and

Second, (4.1)–(4.3) imply in a straightforward way that ˇ ˇ N ˇ ˇ @ 1X 1 @ ˇ D svold 1 .Fi .t// D lim Vd .t; s/ˇ Vd .t; s/ˇˇ @s 2 i D1 ri s0 !0C @s sD0 sDs0

(4.4)

! ˇ ˇ N X ˇ ˇ 1 @ 1 @ d D svold 1 .Fi .t// D lim Vd .t; s/ˇˇ resp:; V .t; s/ˇˇ : (4.5) @s 2 ri s0 !0C @s sD0 sDs0 i D1

Thus, (4.4) and (4.5) together with Lemma 4.1.2 finish the proof of Theorem 3.3.2.

4.2 Proof of Theorem 3.3.3: Codimension Two Volume

71

4.2 Proof of Theorem 3.3.3: Codimension Two Volume Theorem 3.3.3. Let the centers of the closed d -dimensional balls Bd Œpi ; ri , 1 i N lie in the .d 2/-dimensional affine subspace L of Ed ; d S3. If Fi stands d for the contribution of the i th ball to the boundary of the union N i D1 B Œpi ; ri TN (resp., of the intersection i D1 Bd Œpi ; ri ), then vold 2

!

N [

d 2

B

Œpi ; ri D

N \

d 2

i D1

resp:; vold 2

i D1

!

B

1 X 1 svold 1 .Fi / 2 1i N ri

! 1 X 1 Œpi ; ri D svold 1 .Fi / ; 2 1i N ri

where Bd 2 Œpi ; ri D Bd Œpi ; ri \ L; 1 i N . We start our proof with the following volume formula from calculus, which is based on cylindrical shells. Lemma 4.2.1. Let X be a compact measurable set in Ed ; d 3 that is a of revolution aboutEd 2 . In other words the orthogonal projection of X \ ˚solid d 2 E .s cos ; s sin / onto Ed 2 is a measurable set X.s/ independent of . Then Z 1 vold .X / D .2s/vold 2 .X.s// ds: 0

By assumption the centers of the closed d -dimensional balls Bd Œpi ; ri , 1 i N lie in the .d 2/-dimensional affine subspace L of Ed . Now, recall the construction of the following (truncated) Voronoi cells. Vi .d / D fx 2 Ed j for all j; kx pi k2 ri2 kx pj k2 rj2 g; Vi .d / D fx 2 Ed j for all j; kx pi k2 ri2 kx pj k2 rj2 g: The set Vi .d / (resp., Vi .d /) is called the nearest (resp., farthest) point Voronoi cell of the point pi in Ed . Then we restrict each of these sets as follows: Vi .ri ; d / D Vi \ Bd Œpi ; ri ; Vi .ri ; d / D Vi \ Bd Œpi ; ri :

72

4 Proofs on Contractions of Sphere Arrangements

We call the set Vi .ri ; d / (resp., Vi .ri ; d /) the nearest (resp., farthest) point truncated Voronoi cell of the point pi in Ed . As the point configuration p D .p1 ; p2 ; : : : ; pN / lies in the .d 2/-dimensional affine subspace L Ed and as without loss of generality we may assume that L D Ed 2 , therefore one can introduce the relevant .d 2/-dimensional truncated Voronoi cells Vi .ri ; d 2/ and Vi .ri ; d 2/ in a straightforward way. We are especially interested in the relation of the volume of Vi .ri ; d 2/ and Vi .ri ; d 2/ in Ed 2 to the volume of the corresponding truncated Voronoi cells Vi .ri ; d / and Vi .ri ; d / in Ed . Lemma 4.2.2. We have that Z

ri

vold .Vi .ri ; d // D

.2s/vold 2 .Vi .s; d 2// ds;

0

and vold Vi .ri ; d / D

Z

ri

.2s/vold 2 Vi .s; d 2/ ds:

0

Proof. It is clear, in both cases, that Vi .ri ; d / and Vi .ri ; d / are compact measurable d 2 sets of revolution (about E ). Note that the orthogonal projection of Bd Œpi ; ri \ ˚ d 2 E .s cos ; s sin / onto Ed 2 is the .d 2/-dimensional ball of radius q ri2 s 2 centered at pi . Thus, by Lemma 4.2.1 we have that Z

ri

vold .Vi .ri ; d // D 0

q .2s/vold 2 Vi ri2 s 2 ; d 2 ds :

q But if we make the change of variable u D ri2 s 2 , we get the desired integral. A similar calculation works for vold Vi .ri ; d / . The following is an immediate corollary of Lemma 4.2.2. Corollary 4.2.3. We have that ˇ ˇ d vold .Vi .r; d // ˇˇ D 2 ri vold 2 .Vi .ri ; d 2// ; dr rDri and ˇ i ˇ d vold V .r; d / ˇˇ D 2 ri vold 2 Vi .ri ; d 2/ : dr rDri Moreover, it is clearSthat if Fi stands for the contribution of the i th ball to the d boundary of the union N i D1 B Œpi ; ri , then

4.3 Proof of Theorem 3.3.4: The Leapfrog Lemma

73

ˇ ˇ d vold .Vi .r; d // ˇˇ svold 1 .Fi / D : dr rDri

(4.6)

Similarly,Tif Fi denotes the contribution of the i th ball to the boundary of the d intersection N i D1 B Œpi ; ri , then ˇ i ˇ d svold 1 .Fi / D vold V .r; d / ˇˇ : dr rDri

(4.7)

Finally, it is obvious that

vold 2

N [

! d 2

B

Œpi ; ri D

i D1

vold 2

i D1

vold 2 .Vi .ri ; d 2// ;

(4.8)

vold 2 Vi .ri ; d 2/ :

(4.9)

i D1

and N \

N X

! d 2

B

Œpi ; ri D

N X i D1

Thus, Corollary 4.2.3 and (4.6), (4.8) (resp., (4.7), (4.9)) finish the proof of Theorem 3.3.3.

4.3 Proof of Theorem 3.3.4: The Leapfrog Lemma Theorem 3.3.4. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed ; d 1, then there is an analytic contraction p.t/ D .p1 .t/; : : : ; pN .t//; 0 t 1 in E2d such that p.0/ D p and p.1/ D q. Actually, we are going to prove the following even stronger statement. For more information on the background of this theorem we refer the interested reader to [48]. Theorem 4.3.1. Suppose that p and q are two configurations in Ed ; d 1. Then the following is a continuous motion p.t/ D .p1 .t/; : : : ; pN .t// in E2d , that is analytic in t, such that p.0/ D p, p.1/ D q and for 0 t 1, kpi .t/ pj .t/k is monotone: pi qi pi qi pi C qi C .cos t / ; .sin t/ ; 1 i < j N: pi .t/ D 2 2 2

74

4 Proofs on Contractions of Sphere Arrangements

Proof. We calculate: 4kpi .t/ pj .t/k2 D k.pi pj / .qi qj /k2 Ck.pi pj / C .qi qj /k2 C 2.cos t /.kpi pj k2 kqi qj k2 / : This function is monotone, as required.

4.4 Proof of Theorem 3.4.1 Theorem 3.4.1. If a finite set of closed d -dimensional balls of radius 2 (i.e., of closed hemispheres) in the d -dimensional spherical space Sd ; d 2 is rearranged so that the (spherical) distance between each pair of centers does not increase, then the (spherical) d -dimensional volume of the intersection does not decrease and the (spherical) d -dimensional volume of the union does not increase.

4.4.1 The Spherical Leapfrog Lemma As usual, let Sd ; d 2 denote the unit sphere centered at the origin o in Ed C1 , and let X.p/ be a finite intersection of closed balls of radius 2 (i.e., of closed hemispheres) in Sd whose configuration of centers is p D .p1 ; : : : ; pN /. We say that another configuration q D .q1 ; : : : ; qN / is a contraction of p if, for all 1 i < j N , the spherical distance between pi and pj is not less than the spherical distance between qi and qj . We denote the d -dimensional spherical volume measure by Svold ./. Thus, Theorem 3.4.1, that we need to prove, can be phrased as follows: if q is a configuration in Sd that is a contraction of the configuration p, then Svold .X.p// Svold .X.q// :

(4.10)

We note that the part of Theorem 3.4.1 on the union of closed hemispheres is a simple set-theoretic consequence of (4.10). Next, we recall Theorem 4.3.1, which we like to call the (Euclidean) Leapfrog Lemma [48]. We need to apply this to a sphere, rather than Euclidean space. Here we consider the unit spheres Sd Sd C1 Sd C2 in such a way that each Sd is the set of points that are a unit distance from the origin o in Ed C1 . So we need the following. Corollary 4.4.1. Suppose that p and q are two configurations in Sd . Then there is a monotone analytic motion from p to q in S2d C1 .

4.4 Proof of Theorem 3.4.1

75

Proof. Apply Theorem 4.3.1 to each configuration p and q with o as an additional configuration point for each. So for each t, the configuration p.t/ D .p1 .t/; : : : ; pN .t// lies at a unit distance from o in E2d C2 , which is just S2d C1 .

4.4.2 Smooth Contractions via Schl¨afli’s Differential Formula We look at the case when there is a smooth motion p.t/ of the configuration p in Sd . More precisely we consider the family X.t/ D X.p.t// of convex spherical d polytopes in Sd having the same combinatorial face structure with facet hyperplanes being differentiable in the parameter t. The following classical theorem of Schl¨afli (see, e.g., [139]) describes how the volume of X.t/ changes as a function of its dihedral angles and the volume of its .d 2/-dimensional faces. Lemma 4.4.2. For each .d 2/-face Fij .t/ of the convex spherical d -polytope X.t/ in Sd let ˛ij .t/ represent the (inner) dihedral angle between the two facets Fi .t/ and Fj .t/ meeting at Fij .t/. Then the following holds. d d 1 X Svold .X.t// D Svold 2 Fij .t/ ˛ij .t/; dt d 1 F dt ij

to be summed over all .d 2/-faces. Corollary 4.4.3. Let q be a configuration in Sd with a differentiable contraction p.t/ in t of the configuration p in Sd and assume that the convex spherical d -polytopes X.t/ D X.p.t// of Sd have the same combinatorial face structure. Then d Svold .X.t// 0 : dt Proof. As the spherical distance between pi .t/ and pj .t/ is decreasing, the derivad ˛ij .t/ 0. The result then follows from Lemma 4.4.2. tive of the dihedral angle dt

4.4.3 From Higher- to Lower-Dimensional Spherical Volume The last piece of information that we need before we get to the proof of Theorem 3.4.1 is a way of relating higher-dimensional spherical volumes to lowerdimensional ones. Let X be any integrable set in Sn . Recall that we regard X Sn D Sn fog EnC1 EkC1 :

76

4 Proofs on Contractions of Sphere Arrangements

Regard fog Sk EnC1 EkC1 : Let X Sk be the subset of SnCkC1 consisting of the union of the geodesic arcs from each point of X to each point of fog Sk . (So, in particular, Sn Sk D SnCkC1 ). Lemma 4.4.4. For any integrable subset X of Sn , nCkC1 SvolnCkC1 X Sk D Svoln .X / ; n where n D Svoln .Sn /, nCkC1 D SvolnCkC1 Sn Sk D SvolnCkC1 SnCkC1 . Proof. Since the operation (a kind of spherical join) is associative, we only need to consider the case when k D 0. Regard fog S0 D S0 D fn; sg, the north pole and the south pole of SnC1 . We use polar coordinates centered atn to calculate the .n C 1/-dimensional volume of X S0 . Let X.z/ D .X S0 / \ EnC1 fzg , and let be the angle that a point in SnC1 makes with n, the north pole in S nC1 . So z D z. / D cos . Then the spherical volume element for Sn .z/ D SnC1 \ EnC1 fzg is dVn .z/ D .sinn /dVn .0/ because Sn .z/ is obtained from Sn .0/ by a dilation by sin . Then Z SvolnC1 X S0 D dVn .z/d

(4.11) X S0

Z

Z

Z

dVn .z/d D

D 0

X .z. //

.sinn /Vn .X /d

(4.12)

0

Z

D Svoln .X /

.sinn /d D Svoln .X / 0

nC1 ; n

(4.13)

where (4.13) can be seen by taking X D Sn , or by performing the integral explicitly.

4.4.4 Putting Pieces Together Now, we are ready for the proof of Theorem 3.4.1. Let the configuration q D .q1 ; : : : ; qN / be a contraction of the configuration p D .p1 ; : : : ; pN / in Sd . By Corollary 4.4.1, there is an analytic motion p.t/, in S2d C1 for 0 t 1, where p.0/ D p, and p.1/ D q, and all the pairwise distances between the points of p.t/ decrease in t.

4.5 Proof of Theorem 3.4.5

77

Without loss of generality we may assume that X d .p.0// WD X.p.0// is a convex spherical d -polytope in Sd . Since p.t/ is analytic in t, the intersection X 2d C1 .p.t// of the (closed) hemispheres centered at the points of the configuration p.t/ in S2d C1 is a convex spherical .2d C 1/-polytope with a constant combinatorial structure, except for a finite number of points in the interval Œ0; 1. By Corollary 4.4.3, Svol2d C1 X 2d C1 .p.t// is monotone increasing in t. Recall that X d .p/ and X d .q/ are the intersections of the (closed) hemispheres centered at the points of p and q in Sd . From the definition of the spherical join , X d .p/ Sd D X 2d C1 .p/ D X 2d C1 .p.0// X d .q/ Sd D X 2d C1 .q/ D X 2d C1 .p.1//: Hence, by Lemma 4.4.4, Svold X d .p/ D

d 2d C1

d

Svol2d C1 X 2d C1 .p.0//

2d C1 2d C1 Svol2d C1 X .p.1// D Svold X d .q/ :

This finishes the proof of Theorem 3.4.1.

4.5 Proof of Theorem 3.4.5 Theorem 3.4.5. Let P and Q be nonobtuse-angled compact convex polyhedra of the same simple combinatorial type in H3 . If each inner dihedral angle of Q is at least as large as the corresponding inner dihedral angle of P, then the volume of P is at least as large as the volume of Q.

4.5.1 Monotonicity of the Volume of Hyperbolic Simplices Case 4.5.1. P and Q are simplices. Let Xn be the spherical, Euclidean, or hyperbolic space Sn ; En , or Hn of constant curvature C1; 0; 1; and of dimension n 2: By an n-dimensional simplex n in Xn we mean a compact subset with nonempty interior which can be expressed as an intersection of n C 1 closed halfspaces. (In the case of spherical space we require that n lies on an open hemisphere.) Let F0 ; F1 ; : : : ; Fn be the .n 1/-dimensional faces of the simplex n . Each .n 2/-dimensional face can be described uniquely as an intersection Fij D Fi \ Fj : We identify the collection of all inner dihedral

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4 Proofs on Contractions of Sphere Arrangements

angles of the simplex n with the symmetric matrix ˛ D Œ˛ij , where ˛ij is the inner dihedral angle between Fi and Fj for i ¤ j; and where the diagonal entries ˛i i are set equal to by definition.

Definition 4.5.2. The Gram matrix G.n / D gij .n / of the simplex n Xn is the .n C 1/ .n C 1/ symmetric matrix defined by gij .n / D cos ˛ij . Note that all diagonal entries gi i .n / are equal to one. Definition 4.5.3. Let n WD fG.n / j n is an n-dimensional simplex is Sn g ; GC

G0n WD fG.n / j n is an n-dimensional simplex in En g ; Gn WD fG.n / j n is an n-dimensional simplex in Hn g ;

and

n [ G0n [ Gn : G n WD GC n The following lemma summarizes some of the major properties of the sets GC ; n n G , and G that have been studied on several occasions including the papers of Coxeter [82], Milnor [147], and Vinberg [178].

G0n ;

Lemma 4.5.4. (1) The determinant of G.n / is either positive or zero or negative depending on whether the simplex n is spherical or Euclidean or hyperbolic. (2) G n is an open convex set in RN with N D n.nC1/ : (Note that the affine 2 space consisting of all symmetric unidiagonal .n C 1/ .n C 1/ matrices has dimension N D n.nC1/ :/ 2 (3) G0n is an .N 1/-dimensional topological cell that cuts G n into two open n subcells GC and Gn : n n (4) GC (resp., GC [ G0n / is an open convex (resp., closed convex) set in RN : We need the following property for our proof of Theorem 3.4.5 that seems to be n n a new property of GC (resp., GC [ G0n / not yet mentioned in the literature. It is useful to introduce the notations RN <0 WD f.x1 ; x2 ; : : : ; xN / j xi < 0 for all 1 i N g ; and RN 0 WD f.x1 ; x2 ; : : : ; xN / j xi 0 for all 1 i N g : n n n N Lemma 4.5.5. GC \ RN / is a convex corner; that is, if <0 (resp., GC [ G0 \ R 0n n N g D .g1 ; g2 ; : : : ; gN / 2 GC \ R<0 (resp., g 2 GC [ G0n \ RN 0 ), then for any g0 D .g10 ; g20 ; : : : ; gN0 / with g1 g10 < 0; : : : ; gN gN0 < 0 (resp., g1 g10 n 0 n n N 0; : : : ; gN gN0 0/ we have that g0 2 GC \ RN <0 (resp. g 2 GC [ G0 \ R0 ).

4.5 Proof of Theorem 3.4.5

79

Proof. Due to Lemma 4.5.4 it is sufficient to check the claim of Lemma 4.5.5 for n n N the set GC \RN <0 only. Let g D .g1 ; g2 ; : : : ; gN / 2 GC \R<0 : Then it is sufficient to show that for any "1 ; "2 ; : : : ; "N with g1 "1 < 0; g2 "2 < 0; : : : ; gN "N < 0 we have that

(*)

n g1 WD ."1 ; g2 ; : : : ; gN / 2 GC \ RN <0 ; 2 n \ RN g WD .g1 ; "2 ; g3 ; : : : ; gN / 2 GC <0 ; :: : n \ RN gN WD .g1 ; : : : ; gN 1 ; "N / 2 GC <0 :

n Namely, it is easy to see that (*) and the convexity of GC \ RN <0 imply that n N GC \ R<0 is indeed a convex corner. Although it is not needed here, for the sake of n completeness we note that the origin of RN is in fact, an interior point of GC . n n n Let be the n-dimensional simplex of S whose Gram matrix G. / D

n gij . / corresponds to g D .g1 ; g2 ; : : : ; gN /; that is,

.g1 ; g2 ; : : : ; gN / D . cos ˛01 ; : : : ; cos ˛0n ; cos ˛12 ; : : : ; cos ˛.n1/n /: n As g 2 GC \ RN <0 we have that 0 < ˛01 < 2 ; 0 < ˛02 < 2 ; : : : ; 0 < ˛0n < 2 ; 0 < 1 ˛12 < 2 ; : : : ; 0 < ˛.n1/n < 2 . In order to show that g D ."1 ; g2 ; : : : gN / 2 n n n GC \ RN <0 we have to show the existence of an n-dimensional simplex 1 of S with dihedral angles

arccos."1 /; ˛02 ; : : : ; ˛0n ; ˛12 ; : : : ; ˛.n1/n : (As the task left for the remaining parts of (*) is the same we do not give details of that here.) We show the existence of n1 via polarity. Let n D fx 2 Sn j x y 0 for all y 2 n g be the spherical polar of n ; where x y denotes the standard inner product of the unit vectors x and y. As is well known, n is an n-dimensional simplex of Sn with edgelength ˛01 ; ˛02 ; : : : ; ˛0n ; ˛12 ; : : : ; ˛.n1/n each being larger than 2 . Let F be the .n 2/dimensional face of n disjoint from the edge of length ˛01 of n : Let v0 and v1 be the endpoints of the edge of length ˛01 of n . By assumption 2 < arccos."1 / ˛01 < . Now, rotate v1 towards v0 about the .n 2/dimensional greatsphere Sn2 of F in Sn until the rotated image vN 1 of v1 becomes a point of the .n 1/-dimensional greatsphere Sn1 of the facet of n disjoint from v1 . Obviously, the above rotation about Sn2 decreases the (spherical) distance v0 v1 in a continuous way. We claim via continuity that there is a rotated image say, v01 of v1 such that the spherical distance v0 v01 is equal to arccos."1 /. Namely, the nC1 points formed by v0 ; vN 1 and the vertices of F all belong to an open hemisphere of Sn1 with the property that all pairwise spherical distances different from v0 vN 1 are larger than 2 . (Here we assume that v0 and vN 1 are distinct because if they coincide, then the existence of v01 is trivial.) But, then a theorem of Davenport and Haj´os [88]

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4 Proofs on Contractions of Sphere Arrangements

implies that v0 vN 1 2 and so, the existence of v01 follows. Thus, the spherical polar of the n-dimensional simplex of Sn spanned by v0 ; v01 and F gives us n1 . This completes the proof of Lemma 4.5.5. Now, we are in a position to show that Gn \ RN 0 is monotone-path connected. Lemma 4.5.6. Gn \RN 0 is monotone-path connected in the following strong sense. 0 0 0 n N If g D .g1 ; : : : ; gN / 2 Gn \ RN 0 and g D .g1 ; : : : ; gN / 2 G \ R0 with 0 0 0 n N g1 g1 ; : : : ; gN gN ; then g C .1 /g 2 G \ R0 for all 0 1. Proof. Lemma 4.5.4 implies that g0 C .1 /g 2 G n for all 0 1 and n so it is sufficient to prove that g0 C .1 /g 62 GC [ G0n for all 0 1. n n n n n As g 62 GC [G0 and GC [G0 is convex (Lemma 4.5.4) moreover GC [ G0n \RN 0 is a convex corner via Lemma 4.5.5, therefore there exists a supporting hyperplane n H in RN that touches GC [ G0n at some point h 2 G0n \ RN 0 and is disjoint from n n g and separates g from G [ G . In fact, again using the convex corner property of C 0 n N n n GC [ G0n \ RN 0 we get that H separates h C R0 from GC [ G0 and therefore N n n 0 H separates g C R0 from GC [ G0 as well. Finally, notice that g 2 g C RN 0 and N n n is disjoint from H and therefore g C R is disjoint from G [ G g C RN 0 0 C 0 . This finishes the proof of Lemma 4.5.6. Now, we are ready to give a proof of the following volume monotonicity property of hyperbolic simplices. Theorem 4.5.7. Let P and Q be nonobtuse-angled n-dimensional hyperbolic simplices. If each inner dihedral angle of Q is at least as large as the corresponding inner dihedral angle of P, then the n-dimensional hyperbolic volume of P is at least as large as that of Q. Proof. By moving to the space of Gram matrices of n-dimensional hyperbolic simplices and then applying Lemma 4.5.6 we get that there exists a smooth one-parameter family P.t/; 0 t 1 of nonobtuse-angled n-dimensional hyperbolic simplices with the property that P.0/ D P and P.1/ D Q; moreover, if ˛01 .t/; ˛02 .t/; : : : ; ˛0n .t/; ˛12 .t/; : : : ; ˛.n1/n .t/ denote the inner dihedral angles of P.t/; then ˛ij .t/ is a monotone increasing function of t for all 0 i < j n: Now, Schl¨afli’s classical differential formula (see, e.g., [139]) yields that X d 1 d Hvoln .P.t// D Hvoln2 Fij .t/ ˛ij .t/; dt n 1 0i <j n dt

(4.14)

where Fij .t/ denotes the .n 2/-dimensional face of P.t/ on which the dihedral angle ˛ij .t/ sits and Hvoln ./; Hvoln2 ./ refer to the corresponding dimend ˛ij .t/ 0, (4.14) implies that sional hyperbolic volume measures. Thus, as dt d Hvol .P.t// 0 and so, indeed Hvol .P.0// Hvoln .P.1//, finishing the proof n n dt of Theorem 4.5.7.

4.5 Proof of Theorem 3.4.5

81

4.5.2 Smooth One-Parameter Family of Hyperbolic Polyhedra Case 4.5.8. The combinatorial type of P and Q is different from that of a tetrahedron. First, recall the following classical theorem of Andreev [8]. Theorem 4.5.9. A nonobtuse-angled compact convex polyhedron of a given simple combinatorial type, different from that of a tetrahedron and having given inner dihedral angles exists in H3 if and only if the following conditions are satisfied: (1) If three faces meet at a vertex, then the sum of the inner dihedral angles between them is larger than ; (2) If three faces are pairwise adjacent but not concurrent, then the sum of the inner dihedral angles between them is smaller than ; (3) If four faces are cyclically adjacent, then at least one of the dihedral angles between them is different from 2 ; (4) (For triangular prism only) one of the angles formed by the lateral faces with the bases must be different from 2 . Second, observe that the Andreev theorem implies that the space of the inner dihedral angles of nonobtuse-angled compact convex polyhedra of a given simple combinatorial type different from that of a tetrahedron in H3 is a convex set. As a result we get that if P and Q are given as in Theorem 3.4.5 and are different from a tetrahedron, then there exists a smooth one-parameter family P.t/; 0 t 1 of nonobtuse-angled compact convex polyhedra of the same simple combinatorial type as of P and Q with the property that P.0/ D P and P.1/ D Q; moreover, if ˛E .t/ denotes the inner dihedral angle of P.t/ which sits over the edge corresponding to the edge E of P, then ˛E .t/ is a monotone increasing function of t for all edges E of P. Applying Schl¨afli’s differential formula to the smooth one-parameter family P.t/ we get that d 1X d Hvol3 .P.t// D Hlength .Et / ˛E .t/; dt 2 E dt

(4.15)

where Et denotes the edge of P.t/ corresponding to the edge E of P and E (resp., d Et ) runs over all edges of P (resp., P.t/). Hence, as dt ˛E .t/ 0, (4.15) implies d that dt Hvol3 .P.t// 0 and so, indeed Hvol3 .P.0// Hvol3 .P.1//, completing the proof of Theorem 3.4.5.

Chapter 5

Ball-Polyhedra and Spindle Convex Bodies

Abstract In this chapter, we introduce an extension of the theory of convex polyhedral sets (resp., convex bodies) to the family of intersections of finitely many congruent balls, called ball-polyhedra (resp., to the family of intersections of not necessarily finitely many congruent balls, called spindle convex bodies). The basic properties of ball-polyhedra (resp., spindle convex bodies) that are discussed here include separation and support properties for spindle convex bodies, a Carath´eodory-type theorem for spindle convex hulls, and an Euler–Poincar´etype formula for ball-polyhedra in d -dimensional Euclidean space. In addition, we discuss (generalized) billiard trajectories in disk-polygons and an analogue of Blaschke–Lebesgue theorem for disk-polygons. Furthermore, we investigate the problem of characterizing the edge-graphs of ball-polyhedra in Euclidean 3-space. Another topic, discussed in more details, is on global and local rigidity of ball-polyhedra in Euclidean 3-space. Finally, we investigate the status of the longstanding illumination conjecture of V. G. Boltyanski and H. Hadwiger from 1960 for ball-polyhedra (resp., spindle convex bodies) in Euclidean d -space.

5.1 Intersections of Congruent Balls The previous sections indicate a good deal of geometry on unions and intersections of balls that is worthwhile studying. In particular, when we restrict our attention to intersections of balls the underlying convexity suggests a broad spectrum of new analytic and combinatorial results. Based on our interests in discrete geometry we pay special attention to intersections with non-empty interiors of finitely many congruent closed d -dimensional balls in Ed . In fact, one may assume that the congruent d -dimensional balls in question are of unit radius; that is, they are unit balls of Ed . Also, it is natural to assume that removing any of the unit balls defining the intersection in question yields the intersection of the remaining unit balls becoming a larger set. In other words, whenever we take an intersection of finitely many unit balls we always assume that it is generated by a reduced family K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 5, © Springer International Publishing Switzerland 2013

83

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5 Ball-Polyhedra and Spindle Convex Bodies

Fig. 5.1 A disk-triangle in E2

of unit balls. If d D 2, then we call the sets in question disk-polygons and for d 3 they are called ball-polyhedra (Fig. 5.1). In general, due to the underlying convexity, intersections with non-empty interiors of not necessarily finitely many unit balls in Ed are called spindle convex bodies.

5.2 Disk-Polygons Revisited Billiards have been around for quite some time in mathematics and have generated a great deal of research. See, for example, the recent elegant book [174] of Tabachnikov as well as the delightful book [166] of Schoenberg from several years back. For our purposes it seems natural to define billiard trajectories in the following way. This introduces a larger class of polygonal paths for billiard trajectories than the traditional definition widely used in the literature (see [174]). Let C be an arbitrary convex body in the d -dimensional Euclidean space Ed , d 2 that is a compact convex set with non-empty interior in Ed. Then we say that the closed polygonal path P (possibly with self-intersections) is a generalized billiard trajectory of C if all the vertices of P lie on the boundary of C and if each of the (inner) angle bisectors of P between two consecutive sides of P is perpendicular to a supporting hyperplane of C passing through the corresponding vertex of P. If P has n sides, then we say that P is an n-periodic generalized billiard trajectory in C. Note that our definition of generalized billiard trajectories coincides with the traditional definition of billiard trajectories whenever the billiard table has no corner points. It seems that the paper [47] was among the first suggesting a detailed study of generalized billiard trajectories in convex domains. For some analogue higherdimensional investigations we refer the interested reader to the recent paper [106]. Generalized billiard trajectories have the following fundamental property proved by the author and D. Bezdek in the very recent paper [46]. Theorem 5.2.1. Let C be a convex body in Ed , d 2. Then C possesses at least one shortest generalized billiard trajectory; moreover, any of the shortest generalized billiard trajectories in C is of period at most d C 1. For the sake of completeness we mention that according to the main result of [21] any d -dimensional billiard table with a smooth boundary, but not necessarily

5.2 Disk-Polygons Revisited

85

Fig. 5.2 A Reuleaux triangle and its generalized billiard trajectories

convex, has a k-periodic billiard trajectory with k d C1, which is a closely related result. It was observed in [46] that for the following special family of disk-polygons one can improve the estimate on periods in Theorem 5.2.1. If D is a disk-polygon in E2 having the property that the pairwise distances between the centers of its generating unit disks are at most 1, then we say that D is a fat disk-polygon (Fig. 5.2). Theorem 5.2.2. Let D be a fat disk-polygon in E2 . Then any of the shortest generalized billiard trajectories in D is a 2-periodic one. In the recent paper [106], the following fundamental question is studied that was raised by Zelditch in [180] motivated by applications to inverse spectral problems. In which convex bodies are the shortest periodic billiard trajectories of period 2? It is proved in [106] that any convex body whose inscribed ball touches the boundary of the given convex body at two diametrically opposite points has that property. Theorem 5.2.2 shows that the family of fat disk-polygons possesses the same property as well. According to Birkhoff’s well-known theorem [174] if B is a strictly convex billiard table with smooth boundary (i.e., if the boundary of B is a simple, closed, smooth, and strictly convex curve) in E2 , then for every positive integer N > 1 there exist (at least two) N -periodic billiard trajectories in B. (In fact, here the rotation number of the billiard trajectory in question can be preassigned as well. Also, it is well known that neither the convexity nor smoothness can be removed from the assumptions in order to have the same conclusion. Last but not least, for a higher-dimensional analogue of Birkhoff’s theorem we refer the interested reader to [97].) Billiard tables suitable for Birkhoff’s theorem can be easily constructed from disk-polygons as follows. Take a disk-polygon D in E2 . Then choose a positive " not larger than the inradius of D (which is the radius of the largest circular disk contained in D) and take the union of all circular disks of radius " that lie in D. We call the set obtained in this way the "-rounded disk-polygon of D and denote it by D."/. The following theorem proved in [46] also shows the complexity of the problem of Zelditch [180] on characterizing convex domains whose shortest periodic billiard trajectories are of period 2.

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5 Ball-Polyhedra and Spindle Convex Bodies

Theorem 5.2.3. Let D be a fat disk-polygon in E2 . Then any of the shortest (generalized) billiard trajectories in the "-rounded disk-polygon D."/ is a 2-periodic one for all " > 0 being sufficiently small. Actually, we believe [46] that the following even stronger statement holds. Conjecture 5.2.4. Let D be a fat disk-polygon in E2 . Then any of the shortest (generalized) billiard trajectories in the "-rounded disk-polygon D."/ is a 2-periodic one for all " being at most as large as the inradius of D. Finally we mention the following result that can be obtained as an immediate corollary of Theorem 5.2.2. This might be of independent interest, in particular because it generalizes the result proved in [47] that any closed curve of length at most 1 can be covered by a translate of any convex domain of constant width 12 in the Euclidean plane. As usual if C is a convex domain of the Euclidean plane, then let w.C/ denote the minimal width of C (i.e., the smallest distance between two parallel supporting lines of C). Corollary 5.2.5. Let D be a fat disk-polygon in E2 . Then any closed curve of length at most 2w.D/ in E2 can be covered by a translate of D. It would be interesting to find the higher-dimensional analogues of the 2dimensional results just mentioned in this section. In particular, the following question does not seem to be an easy one. Problem 5.2.6. Let P be a ball-polyhedron in Ed ; d 3 with the property that the pairwise distances between the centers of its generating unit balls are at most 1. Then prove or disprove that any of the shortest generalized billiard trajectories in P is a 2-periodic one. The classical isoperimetric inequality combined with Barbier’s theorem (stating that the perimeter of any convex domain of constant width w is equal to w) implies that the largest area of convex domains of constant width w is the circular disk of diameter w, having the area of 4 w2 (for more details see, for example, [64]). On the other hand, the well-known Blaschke-Lebesgue theorem states that among all convex domains of constant pwidth w, the Reuleaux triangle of width w has the smallest area, namely 12 . 3/w2 . Blaschke [60] and Lebesgue [143] were the first to show this and the succeeding decades have seen other works published on different proofs of that theorem. For a most recent new proof, and for a survey on the state of the art of different proofs of the Blaschke–Lebesgue theorem, see the elegant paper of Harrell [125]. The main goal of this section is to extend the Blaschke–Lebesgue Theorem for disk-polygons. p The disk-polygon D is called a disk-polygon with center parameter t, 0 < t < 3 D 1:732 : : : , if the distance between any two centers of the generating unit disks of D is at most t. Let F .t/ denote the family of all disk-polygons with center parameter t. Let .t/ denote the regular disk-triangle whose three generating p unit disks are centered at the vertices of a regular triangle of side length t, 1 t < 3 D 1:732 : : : . Recall that the inradius r.C/ of a convex domain C in E2 is the radius of

5.2 Disk-Polygons Revisited

87

the largest circular disk lying in C (simply called the incircle of C). The following formulas give the inradius r..t//, the minimal width p w..t//, the area a..t// and the perimeter p..t// of .t/ for all 1 t < 3: r..t// D 1

w..t// D 1

a..t// D 3 arccos

1 2

q

1p 3tI 3

p p 4 C 2t 2 2 3t 4 t 2 I

1p 2 3 p t 1 C 3t t 4 t 2 I 2 4 4 2

t p..t// D 2 6 arcsin : 2 The following theorem has been proved by M. Bezdek [40]. Theorem 5.2.7. Let D 2 F .t/ be an arbitrary disk-polygon with center parameter p t; 1 t < 3. Then r.D/ r..t// and w.D/ w..t//. Moreover, the area of D is at least as large as the area of .t/; that is, a.D/ a..t// with equality if and only if D D .t/. For t D 1 the above area inequality and the well-known fact (see, e.g., [64]) that the family of Reuleaux polygons of width 1 is a dense subset of the family of convex domains of constant width 1, imply the Blaschke–Lebesgue theorem in a straightforward way. In connection with Theorem 5.2.7 we propose to investigate the following related problem, in particular, because an affirmative answer to that question would imply the area inequality of Theorem 5.2.7. Problem 5.2.8. p Let D 2 F .t/ be an arbitrary disk-polygon with center parameter t; 1 t < 3. Prove or disprove that the perimeter of D is at least as large as the perimeter of .t/; that is, p.D/ p..t//: Let C E2 be a convex domain and let > 0 be given. Then, the outer parallel domain C of radius of C is the union of all (closed) circular disks of radii , whose centers belong to C. Recall that a.C / D a.C/ C p.C/ C 2 . Let 0 < t < 1 be given and let R.t/1t denote the outer parallel domain of radius 1 t of a Reuleaux triangle R.t/ of width t. Note that R.t/1t is a convex domain of

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5 Ball-Polyhedra and Spindle Convex Bodies

constant width 2 t and so, Barbier’s theorem [64] implies that its perimeter is equal to p .R.t/1t / D .2 pt/; moreover, it is not hard to check that its area is equal to a .R.t/1t / D 12 . 3/t 2 t C . The following theorem is a natural counterpart of Theorem 5.2.7. Theorem 5.2.9. Let D 2 F .t/ be an arbitrary disk-polygon with center parameter t; 0 < t < 1. Then, the area of D is strictly larger than the area of R.t/1t ; that is, a.D/ > a .R.t/1t / :

5.3 On a Steinitz-Type Problem for Ball-Polyhedra One can represent the boundary of a ball-polyhedron in E3 as the union of vertices, edges, and faces defined in a rather natural way as follows. A boundary point is called a vertex if it belongs to at least three of the closed unit balls defining the ballpolyhedron. A face of the ball-polyhedron is the intersection of one of the generating closed unit balls with the boundary of the ball-polyhedron. Finally, if the intersection of two faces is non-empty, then it is the union of (possibly degenerate) circular arcs. The non-degenerate arcs are called edges of the ball-polyhedron. Obviously, if a ball-polyhedron in E3 is generated by at least three unit balls, then it possesses vertices, edges, and faces. Clearly, the vertices, edges and faces of a ball-polyhedron (including the empty set and the ball-polyhedron itself) are partially ordered by inclusion forming the vertex-edge-face structure of the given ball-polyhedron. It was noted in [59] that the vertex-edge-face structure of a ball-polyhedron is not necessarily a lattice (i.e., a partially ordered set (also called a poset) in which any two elements have a unique supremum (the elements’ least upper bound; called their join) and an infimum (greatest lower bound; called their meet)). Thus, it is natural to define the following fundamental family of ball-polyhedra, introduced in [59] under the name standard ball-polyhedra and investigated in [52] as well without having a particular name for it. Here a ball-polyhedron in E3 is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). This is the case if, and only if, the intersection of any two faces is either empty, or one vertex or one edge, and every two edges share at most one vertex. In this case, we simply call the vertex-edge-face structure in question the face lattice of the standard ball-polyhedron. This definition implies among others that any standard ball-polyhedron of E3 is generated by at least four unit balls. For a number of important properties of ball-polyhedra we refer the interested reader to [52, 59], and [142]. For us a graph is always a non-oriented one that has finitely many vertices and edges. Also, recall that a graph is 3-connected if it has at least four vertices and deleting any two vertices yields a connected graph. Moreover, a graph is called simple if it contains no loops (edges with identical endpoints) and no parallel edges (edges with the same two endpoints). Finally, a graph is planar if it can be drawn

5.4 On Global and Local Rigidity of Ball-Polyhedra

89

in the Euclidean plane without crossing edges. Now, recall that according to the well-known theorem of Steinitz (see pp. 235–244 in [110]) a graph is the edgegraph of some convex polyhedron in E3 if, and only if, it is simple, planar, and 3-connected. As a partial analogue of Steinitz’s theorem for ball-polyhedra the following theorem is proved in [59]. Theorem 5.3.1. The edge-graph of any standard ball-polyhedron in E3 is a simple, planar, and 3-connected graph. Based on that it would be natural to look for an answer to the following question raised in [59]. Problem 5.3.2. Prove or disprove that every simple, planar, and 3-connected graph is the edge-graph of some standard ball-polyhedron in E3 . In connection with the above problem we recall from [59] that if a simple, planar, and 3-connected graph can be realized as the edge-graph of a convex polyhedron in E3 with all faces inscribed in a circle, then it can be realized as the edge-graph of a standard ball-polyhedron in E3 . However, not every simple, planar, and 3-connected graph can be realized as the edge-graph of a convex polyhedron in E3 with all faces having a circumcircle. See pp. 286–287 in [110].

5.4 On Global and Local Rigidity of Ball-Polyhedra We state our rigidity results on ball-polyhedra together with some well-known theorems on convex polyhedra. In fact, those classical theorems on convex polyhedra have motivated our work on ball-polyhedra a great deal. The details are as follows. One of the best known results on convex polyhedra is Cauchy’s celebrated rigidity theorem [73]. (For a recent account on Cauchy’s theorem see Chap. 11 of the mathematical bestseller [1] as well as Theorem 26.6 and the discussion followed in the elegant book [156].) Cauchy’s theorem is often quoted as follows: If two convex polyhedra P and P0 in E3 are combinatorially equivalent with the corresponding faces being congruent, then P is congruent to P0 . It is immediate to note that the analogue of Cauchy’s theorem for ball-polyhedra is a rather obvious statement and so, we do not discuss that here. Next, it is natural to recall Alexandrov’s theorem [5] in particular, because it implies Cauchy’s theorem (see also Theorem 26.8 and the discussion followed in [156]): if P and P0 are combinatorially equivalent convex polyhedra with equal corresponding face angles in E3 , then P and P0 have equal corresponding inner dihedral angles. Somewhat surprisingly, the analogue of Alexandrov’s theorem for ball-polyhedra is not trivial. Still, one can prove it following the ideas of the original proof of Alexandrov’s theorem [5]. This was published in [52] (see Claim 5.1 and the discussion followed). In order to state it properly, we need to recall some additional terminology. To each edge of a ballpolyhedron in E3 we can assign an inner dihedral angle. Namely, take any point p in the relative interior of the edge and take the two unit balls that contain the two

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faces of the ball-polyhedron meeting along that edge. Now, the inner dihedral angle along this edge is the angular measure of the intersection of the two half-spaces supporting the two unit balls at p. The angle in question is obviously independent of the choice of p. Moreover, at each vertex of a face of a ball-polyhedron there is a face angle which is the angular measure of the convex angle formed by the two tangent half-lines of the two edges meeting at the given vertex. Finally, we say that the standard ball-polyhedra P and P0 in E3 are combinatorially equivalent if there is an inclusion (i.e., partial order) preserving bijection between the face lattices of P and P0 . Thus, [52] proves the following analogue of Alexandrov’s theorem for standard ball-polyhedra. Theorem 5.4.1. If P and P0 are two combinatorially equivalent standard ballpolyhedra with equal corresponding face angles in E3 , then P and P0 have equal corresponding inner dihedral angles. An important close relative of Cauchy’s rigidity theorem is Stoker’s theorem [172] (see also Theorem 26.9 and the discussion followed in [156]): if P and P0 are two combinatorially equivalent convex polyhedra with equal corresponding edge lengths and inner dihedral angles in E3 , then P and P0 are congruent. Using the ideas of original proof [172] of Stoker’s theorem, the author [39] succeeded in proving the following analogue of Stoker’s theorem for standard ball-polyhedra. Theorem 5.4.2. If P and P0 are two combinatorially equivalent standard ballpolyhedra with equal corresponding edge lengths and inner dihedral angles in E3 , then P and P0 are congruent. Based on the above mentioned analogue of Alexandrov’s theorem for standard ball-polyhedra, Theorem 5.4.2 implies the following statement in straightforward way. Corollary 5.4.3. If P and P0 are two combinatorially equivalent standard ballpolyhedra with equal corresponding edge lengths and face angles in E3 , then P and P0 are congruent. In order to strengthen the above mentioned analogue of Alexandrov’s theorem for standard ball-polyhedra, we recall the following notion from [52]. We say that the standard ball-polyhedron P in E3 is globally rigid with respect to its face angles (resp., globally rigid with respect to its inner dihedral angles) within the family of standard ball-polyhedra if the following holds. If P0 is another standard ballpolyhedron in E3 whose face lattice is combinatorially equivalent to that of P and whose face angles (resp., inner dihedral angles) are equal to the corresponding face angles (resp., inner dihedral angles) of P, then P0 is congruent to P. Furthermore, a ball-polyhedron of E3 is called simplicial if all its faces are bounded by three edges. It is not hard to see that any simplicial ball-polyhedron is, in fact, a standard one. Now, recall the following theorem proved in [52] (see Theorem 0.2). Theorem 5.4.4. If P is a simplicial ball-polyhedron in E3 , then P is globally rigid with respect to its face angles (within the family of standard ball-polyhedra).

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This raises the following question. Problem 5.4.5. Prove or disprove that every standard ball-polyhedron of E3 is globally rigid with respect to its face angles within the family of standard ballpolyhedra. We do not know whether the condition “standard” in Problem 5.4.5 is necessary. However, if the ball-polyhedron Q fails to be a standard ball-polyhedron because it possesses a pair of faces sharing more than one edge, then Q is flexible (and so, it is not globally rigid) as shown in Sect. 4 of [52]. Based on [39] we give a positive answer to Problem 5.4.5 within the following subfamily of standard ball-polyhedra. In order to define the new family of ballpolyhedra in an elementary way, we first take a ball-polyhedron P in E3 with the property that the center points of its generating unit balls are not on a plane of E3 . (We note that this condition is necessary as well as sufficient for having at least one vertex in the underlying farthest-point Voronoi tiling of the center points of the generating unit balls of P. For more details on farthest-point Voronoi tilings see Sect. 6.4.) Then we label the union of the generating unit balls of P by P[ and call it the flower-polyhedron assigned to P. Next, we say that a sphere of E3 is a circumscribed sphere of the flower-polyhedron P[ if it contains P[ (i.e., bounds a closed ball containing P[ ) and touches some of the unit balls of P[ such that there is no other sphere of E3 touching the same collection of unit balls of P[ and contaning P[ . Finally, we call P a normal ball-polyhedron if the radius of every circumscribed sphere of the flower-polyhedron P[ is less than 2. For the sake of completeness we note that the above definition of normal ball-polyhedra is equivalent to the following one: P is a normal ball-polyhedron if and only if P is a ball-polyhedron in E3 with the property that the non-empty family of the vertices of the underlying farthestpoint Voronoi tiling of the center points of the generating unit balls of P is a subset of the interior of P. (Actually, the latter condition is equivalent to the following one: the distance between any center point of the generating unit balls of P and any of the vertices of the farthest-point Voronoi cell assigned to the center in question is strictly less than one.) In the proof of the following theorem we show that every normal ball-polyhedron is in fact, a standard one. On the other hand, it is easy to see that there are standard ball-polyhedra that are not normal ones. The following construction is a general one however, for the sake of simplicity we introduce it for the case of four unit balls: Take four points in convex and generic position in E3 . Construct the farthest-point Voronoi tiling of the four points in E3 , and let l be the largest distance between a vertex of a Voronoi cell and the corresponding point assigned to the Voronoi cell in question. If 0 < r1 < l < r2 and r1 is sufficiently close to l, then the intersection of the four balls having radii r1 (resp., r2 ) centered around the original four points is a standard (resp., normal) ball-polyhedron apart from the normalization of the radius r1 (resp., r2 ). More importantly, the standard ball-polyhedron obtained in this way is not a normal one. The following theorem is a stronger version of the relevant theorem proved in [39], which is stated here as a corollary. We call them the global rigidity analogues of Alexandrov’s theorem for normal ball-polyhedra.

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Theorem 5.4.6. Every normal ball-polyhedron of E3 is globally rigid with respect to its inner dihedral angles within the family of normal ball-polyhedra. Theorem 5.4.6 combined with Theorem 5.4.1 yields the following Corollary 5.4.7. Every normal ball-polyhedron of E3 is globally rigid with respect to its face angles within the family of normal ball-polyhedra. Theorem 5.4.6 leads to a stronger version of Problem 5.4.5 as follows. Clearly, a positive answer to Problem 5.4.8 would imply a positive answer to Problem 5.4.5 (just use Theorem 5.4.1) , but not necessarily the other way around. Problem 5.4.8. Prove or disprove that every standard ball-polyhedron of E3 is globally rigid with respect to its inner dihedral angles within the family of standard ball-polyhedra. One can regard this problem as an extension of the (still unresolved) conjecture of Stoker [172] according to which for convex polyhedra the face lattice and the inner dihedral angles determine the face angles. For an overview on the status of the Stoker conjecture and in particular, for the breakthrough result of Mazzeo and Montcouquiol on proving the infinitesimal version of the Stoker conjecture see [145]. The following special case of Problem 5.4.8 has already been put forward as a conjecture in [52]. For this we need to recall that a ball-polyhedron is called a simple ball-polyhedron, if at every vertex exactly three edges meet. Now, based on our terminology introduced above the conjecture in question ([52], p. 257) can be phrased as follows. Conjecture 5.4.9. Let P be a simple and standard ball-polyhedron of E3 . Then P is globally rigid with respect to its inner dihedral angles within the family of standard ball-polyhedra. Just very recently the author and Nasz´odi [53] were able to give a proof of the following local version of Conjecture 5.4.9. We say that the standard ballpolyhedron P of E3 is locally rigid with respect to its inner dihedral angles, if there is an " > 0 with the following property. If Q is another standard ball-polyhedron of E3 whose face lattice is isomorphic to that of P and whose inner dihedral angles are equal to the corresponding inner dihedral angles of P such that the corresponding faces of P and Q lie at Hausdorff distance at most " from each other, then P and Q are congruent. Thus, the main theorem of [53] can be phrased as follows. Theorem 5.4.10. Let P be a simple and standard ball-polyhedron of E3 . Then P is locally rigid with respect to its inner dihedral angles. Also, it is natural to say that the standard ball-polyhedron P of E3 is locally rigid with respect to its face angles, if there is an " > 0 with the following property. If Q is another standard ball-polyhedron of E3 whose face lattice is isomorphic to that of P and whose face angles are equal to the corresponding face angles of P such that the corresponding faces of P and Q lie at Hausdorff distance at most " from each other,

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then P and Q are congruent. As according to Theorem 5.4.1 the face lattice and the face angles determine the inner dihedral angles of any standard ball-polyhedron in E3 therefore Theorem 5.4.10 implies the following claim in a straightforward way. Corollary 5.4.11. Let P be a simple and standard ball-polyhedron of E3 . Then P is locally rigid with respect to its face angles.

5.5 Separation and Support for Spindle Convex Bodies The following theorem of Kirchberger is well known (see, e.g., [17]). If A and B are finite (resp., compact) sets in Ed with the property that for any set T A [ B of cardinality at most d C 2 (i.e., with card T d C 2) the two sets A \ T and B \ T can be strictly separated by a hyperplane, then A and B can be strictly separated by a hyperplane. It is shown in [59] that no similar statement holds for separation by unit spheres. However, [59] proves the following analogue of Kirchberger’s theorem for separation by spheres of radius at most one. For this purpose it is convenient to denote the .d 1/-dimensional sphere of Ed centered at the point c and having radius r with S d 1 .c; r/ and say that the sets A Ed and B Ed are strictly separated by S d 1 .c; r/ if both A and B are disjoint from S d 1 .c; r/ and one of them lies in the interior and the other one in the exterior of S d 1 .c; r/. Also, we denote by Bd .c; r/ (resp., Bd Œc; r) the open (resp., closed) ball of radius r centered at the point c in Ed . Thus, we say that the sets A and B are separated by S d 1 .c; r/ in Ed if either A Bd Œc; r and B Ed n Bd .c; r/, or B Bd Œc; r and A Ed n Bd .c; r/. Theorem 5.5.1. Let A; B Ed be finite sets. Then A and B can be strictly separated by a sphere S d 1 .c; r/ with r 1 such that A Bd .c; r/ if and only if the following holds. For every T A [ B with card T d C 2, T \ A and T \ B can be strictly separated by a sphere S d 1 .cT ; rT / with rT 1 such that T \ A Bd .cT ; rT /. It is natural to proceed with some basic support and separation properties of convex sets of special kind that include ball-polyhedra. For this purpose let us recall the following definition from [59]. Let a and b be two points in Ed . If ka bk < 2, then the (closed) spindle of a and b, denoted by Œa; bs , is defined as the union of circular arcs with endpoints a and b that are of radii at least one and are shorter than a semicircle. If ka bk D 2, then Œa; bs WD Bd Œ aCb 2 ; 1. If ka bk > 2, then we d d define Œa; bs to be E . Next, a set C E is called spindle convex if, for any pair of points a; b 2 C, we have that Œa; bs C. Finally, recall that if a closed unit ball Bd Œc; 1 contains a set C Ed and a point x 2 bdC is on S d 1 .c; 1/, then we say that S d 1 .c; 1/ or Bd Œc; 1 supports C at x. (Here, as usual, the boundary of a set X Ed is denoted by bdX .) Theorem 5.5.2. Let A Ed be a compact convex set. Then the following are equivalent.

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(i) A is spindle convex. (ii) A is the intersection of unit balls containing it. (iii) For every boundary point of A, there is a unit ball that supports A at that point. Recall that the interior of a set X Ed is denoted by intX . Theorem 5.5.3. Let C; D Ed be spindle convex sets. Suppose C and D have disjoint relative interiors. Then there is a closed unit ball Bd Œc; 1 such that C Bd Œc; 1 and D Ed n Bd .c; 1/. Furthermore, if C and D have disjoint closures and one, say C, is different from a unit ball, then there is an open unit ball Bd .c; 1/ such that C Bd .c; 1/ and D Ed n Bd Œc; 1.

5.6 A Carath´eodory-Type Theorem for Spindle Convex Hulls In this section we study the spindle convex hull of a set and give analogues of the well-known theorems of Carath´eodory and Steinitz to spindle convexity. Carath´eodory’s theorem (see, e.g., [17]) states that the convex hull of a set X Ed is the union of simplices with vertices in X . Steinitz’s theorem (see, e.g., [17]) is that if a point is in the interior of the convex hull of a set X Ed , then it is also in the interior of the convex hull of at most 2d points of X . This number 2d cannot be reduced as shown by the cross-polytope and its center point. Recall the following definition introduced in [59]. Let X be aTset in Ed . Then the spindle convex hull of X is the set defined by convs X WD fC Ed jX C and C is spindle convex in Ed g. Based on this, we can now phrase the major result of this section proved in [59]. Theorem 5.6.1. Let X Ed be a closed set. (i) If y 2 bd.convs X /, then there is a set fx1 ; x2 ; : : : ; xd g X such that y 2 convs fx1 ; x2 ; : : : ; xd g. (ii) If y 2 int.convs X /, then there is a set fx1 ; x2 ; : : : ; xd C1 g X such that y 2 int.convs fx1 ; x2 ; : : : ; xd C1 g/.

5.7 Illuminating Ball-Polyhedra and Spindle Convex Bodies Recall the Boltyanski–Hadwiger Illumination Conjecture [63, 111]. Let K be a convex body (i.e. a compact convex set with nonempty interior) in the d -dimensional Euclidean space Ed , d 2. According to Boltyanski [63] the direction v 2 Sd 1 (i.e. the unit vector v of Ed ) illuminates the boundary point b of K if the halfline emanating from b having direction vector v intersects the interior of K, where Sd 1 Ed denotes the .d 1/-dimensional unit sphere centered at the origin o of Ed . Furthermore, the directions v1 ; v2 ; : : : ; vn illuminate K if each boundary point of

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K is illuminated by at least one of the directions v1 ; v2 ; : : : ; vn . Finally, the smallest n for which there exist n directions that illuminate K is called the illumination number of K denoted by I.K/. An equivalent but somewhat different looking concept of illumination was introduced by Hadwiger in [111]. There he proposed to use point sources instead of directions for the illumination of convex bodies. Based on these circumstances the following conjecture, that was independently raised by Boltyanski [63] and Hadwiger [111] in 1960, is called the Boltyanski–Hadwiger Illumination Conjecture: The illumination number I.K/ of any convex body K in Ed , is at most 2d and I.K/ D 2d if and only if K is an affine d -cube. This conjecture is proved for d D 2 and it is open for all d 3 despite the large number of partial results presently known. The following, rather basic principle, can be quite useful for estimating the illumination numbers of some convex bodies in particular, in low dimensions. Theorem 5.7.1. Let K Ed , d 3 be a convex body and let r be a positive real number with the property that the Gauss image .F / of any face F of K can be covered by a spherical ball of radius r in Sd 1 . Moreover, assume that there exist N points of Sd 1 with covering radius R satisfying the inequality r C R 2 . Then I.K/ N . Using Theorem 5.7.1 as well as the optimal codes for the covering radii of 4 and 5 points on S2 [104] one can prove the first and the second inequality of the theorem stated below. The third inequality has been proved in [59]. Theorem 5.7.2. Let BŒX be a ball-polyhedron in E3 , which is the intersection of the closed 3-dimensional unit balls centered at the points of X E3 . (i) If the Euclidean diameter diam.X / of X satisfies 0 < diam.X / 0:577, then I.BŒX / D 4; (ii) If diam.X / satisfies 0:577 < diam.X / 0:774, then I.BŒX / 5; (iii) If 0:774 < diam.X / 1, then I.BŒX / 6. By taking a closer look of Schramm’s elegant paper [167] and making the necessary modifications, the author [41] improved somewhat the main estimate of [167], but more importantly he succeeded in extending that estimate to the following family of convex bodies (called the family of fat spindle convex bodies) that is much larger than the family of convex bodies of constant width (playing a central role in [167]) and also includes the family of fat ball-polyhedra. Thus, we have the following theorem proved in [41]. Theorem 5.7.3. Let X Ed , d 3 be an arbitrary compact set with diam.X / 1 and let BŒX be the intersection of the closed d -dimensional unit balls centered at the points of X . Then I.BŒX / < 4

12 3

3

d 2 .3 C ln d /

d2 d2 3 3 3 < 5d 2 .4 C ln d / : 2 2

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12

3

3

d 2 .3 C ln d /

3 d2

< 2d for all d 15. (Moreover, for d p 1:5 C " D .1:224 : : : C every " > 0 if d is sufficiently large, then I.BŒX / < On the one hand, 4

2

"/d .) On the other hand, based on the elegant construction of Kahn and Kalai [135], it is known (see [1]), that if d is sufficiently large, then there exists a finite subset X 00 of f0; 1gd in Ed psuch that any partition of X 00 into parts of smaller diameter requires more than .1:2/ d parts. Let X 0 be the (positive) homothetic copy of X 00 having unit diameter and let X be the (not necessarily unique) convex body of constant width p one containing X 0 . Then it follows via standard arguments that I.BŒX / > .1:2/ d with X D BŒX . The natural question whether there exist fat ball-polyhedra with the same property remains open. Theorems 5.7.2 and 5.7.3 suggest attacking the Boltyanski–Hadwiger Illumination Conjecture by letting 0 < diam.X / < 2 to get arbitrarily close to 2 with circumradius 0 < cr.X / < 1. One of the key steps in the proof of Theorem 5.7.3, presented in the relevant section of this paper, is Lemma 6.10.6. In fact, a better lower bound for Lemma 6.10.6 d could lead to an improvement in the exponential factor 32 2 of Theorem 5.7.3. As the underlying spherical geometry problem of Lemma 6.10.6 might be of independent interest we phrase it in a slightly different but equivalent way and make some comments. In order to do so we recall some standard terminology. By a convex body C in Sd 1 we understand the intersection Sd 1 \Co , where Co stands for a linefree d -dimensional closed convex cone with apex o in Ed . We denote by KSd 1 the space of all convex bodies in Sd 1 , equipped with the Hausdorff metric. L Sd 1 is called a lune of Sd 1 if it is the intersection of two (distinct) closed hemispheres of Sd 1 having nonempty interior. The width of L is simply the angular measure of the dihedral angle pos.L/, where pos./ refers to the positive hull of the corresponding set in Ed . The minimal width Swidth.C/ of C 2 KSd 1 is the smallest width of the lunes that contain C. Also, we say that C 2 KSd 1 is a convex body of constant width w if w D Swidth.C/ D Sdiam.C/, where Sdiam./ refers to the spherical diameter of the corresponding set in Sd 1 . For C 2 KSd 1 the polar body C of C is defined by C WD fx 2 Sd 1 j hx; ci 0 for all c 2 Cg; where h; i refers to the canonical inner product in Ed . (The induced canonical Euclidean norm on Ed will be denoted by k k.) Clearly, C 2 KSd 1 . Now, the problem studied in Lemma 6.10.6 is equivalent to the following. (Actually, for a proof of the equivalence one can use the theorem proved in [92] according to which any subset of Sd 1 having spherical diameter 0 < w 2 can be covered by a convex body of constant width w in Sd 1 . Moreover, the polar body of such a convex body is of constant width 2 w < .) Let the positive real 2 w < and the positive integer d 3 be given. Then find the minimum volume convex body of constant width w in Sd 1 . In fact, the question makes sense to ask for all

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0 < w < . Thus, we have arrived at the following quite basic volume problem, whose Euclidean counterpart has been much better studied and is also better known (see for example [18]). Problem 5.7.4. For 0 < w < and d 3 find the minimum volume convex body of constant width w in Sd 1 . Problem 5.7.4 has been solved by Blaschke on S2 (i.e., for d D 3) in [61]. Blaschke’s theorem [61] can be summarized as follows. Among all convex domains of constant width 0 < w 2 on S2 , the Reuleaux triangle of constant width w has the smallest area. Moreover, among all convex domains of constant width < w < of S2 the smallest area belongs to the one which is obtained as the 2 outer parallel domain of radius w 2 of the Reuleaux triangle of width w . The author [41] proved the following partial extension of this theorem of Blaschke to S3 . Theorem 5.7.5. Let 0 < w < be given. Then the volume of the convex body C of constant width w in S3 is minimal among all convex bodies of constant width w of S3 if and only if the polar body C of constant width w has minimal volume among all convex bodies of constant width w of S3 . Thus, Theorem 5.7.5 implies that if d D 4, then it is sufficient to investigate Problem 5.7.4 for convex bodies of constant width 0 < w 2 in S3 . The question whether a statement similar to Theorem 5.7.5 holds in spherical spaces of dimensions 4 and higher remains open. Finally, we wish to call the reader’s attention to the following special case of Problem 5.7.4 that is strikingly simple to phrase in all spherical dimensions. Conjecture 5.7.6. Among all convex bodies of constant width 2 in Sd 1 , d 4, the .d 1/-dimensional regular simplex of edge length 2 has the smallest volume.

5.8 An Euler–Poincar´e-Type Formula for Ball-Polyhedra The main result of this section, published in [59], is an Euler–Poincar´e-type formula for a large family of ball-polyhedra of Ed called standard ball-polyhedra. This family of ball-polyhedra is an extension of the relevant 3-dimensional family of standard ball-polyhedra already discussed in previous sections. The details are as follows. Let S l .p; r/ be a sphere of Ed . The intersection of S l .p; r/ with an affine subspace of Ed that passes through p is called a great-sphere of S l .p; r/. Note that S l .p; r/ is a great-sphere of itself. Moreover, any great-sphere is itself a sphere. Next, let P Ed be a ball-polyhedron with the family of generating balls Bd Œx1 ; 1; : : : ; Bd Œxk ; 1 (meaning that P D \kiD1 Bd Œxi ; 1). Also, recall that by definition removing any of the balls in question yields that the intersection of the remaining balls becomes a set larger than P. The boundary of a generating ball of

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P is called a generating sphere of P. A supporting sphere S l .p; r/ of P is a sphere of dimension l, where 0 l d 1, which can be obtained as an intersection of some of the generating spheres of P such that P \ S l .p; r/ ¤ ;. Note that the intersection of finitely many spheres in Ed is either empty, or a sphere, or a point. In the same way that the faces of a convex polytope can be described in terms of supporting affine subspaces, we describe the faces of a certain class of ball-polyhedra in terms of supporting spheres. Thus, let P be a d -dimensional ballpolyhedron. We say that P is standard if for any supporting sphere S l .p; r/ of P the intersection F WD P \ S l .p; r/ is homeomorphic to a closed Euclidean ball of some dimension. We call F a face of P; the dimension of F is the dimension of the ball to which F is homeomorphic. If the dimension is 0; 1, or d 1, then we call the face a vertex, an edge, or a facet, respectively. Note that the dimension of F is independent of the choice of the supporting sphere containing F . The following theorem has been proved in [59], the last part of which is the desired Euler–Poincar´e formula for standard ball-polyhedra. Theorem 5.8.1. Let be the set containing all faces of a standard ball-polyhedron P Ed and the empty set and P itself. Then is a finite bounded lattice with respect to ordering by inclusion. The atoms of are the vertices of P and is atomic; that is, for every element F 2 with F ¤ ; there is a vertex v of P such that v 2 F . Moreover, P has k-dimensional faces for every 0 k d 1 and P is the spindle convex hull of its .d 2/-dimensional faces. Furthermore, no standard ball-polyhedron in Ed is the spindle convex hull of its .d 3/-dimensional faces. d C1 Finally, D Pd 1 if fii .P/ denotes the number of i -dimensional faces of P, then 1C.1/ i D0 .1/ fi .P/.

Chapter 6

Proofs on Ball-Polyhedra and Spindle Convex Bodies

Abstract First, we prove that any of the shortest generalized billiard trajectories in an arbitrary convex body of Euclidean d -space is of period of at most d C1. Second, we prove an analogue of Stoker’s rigidity theorem for standard ball-polyhedra. Third, we give a proof of the global rigidity analogue of Alexandrov’s theorem for normal ball-polyhedra. Next, we show that every simple and standard ballpolyhedron of Euclidean 3-space is locally rigid with respect to its inner dihedral angles (resp., face angles). Then we prove some basic separation and support properties for spindle convex bodies as well as give a proof of a Charath´eodory-type theorem for spindle convex hulls. Furthermore, we prove an Euler-Poincar´e-type formula for standard ball-polyhedra in Euclidean d -space. Finally, we give a proof of the long-standing Boltyanski-Hadwiger illumination conjecture for fat spindle convex bodies in Euclidean dimensions greater than or equal to 15.

6.1 Proof of Theorem 5.2.1 Theorem 5.2.1. Let C be a convex body in Ed , d 2. Then C possesses at least one shortest generalized billiard trajectory; moreover, any of the shortest generalized billiard trajectories in C is of period at most d C 1.

6.1.1 On Translates of the Interior of a Convex Body We start with the following rather natural statement that can be proved easily with the help of Helly’s theorem [65]. Lemma 6.1.1. Let F be a finite set of at least d C 1 points and C be a convex set in Ed ; d 2. Then C has a translate that covers F if and only if every d C 1 points of F can be covered by a translate of C. K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 6, © Springer International Publishing Switzerland 2013

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Proof. For each point p 2 F let Cp denote the set of all translation vectors in Ed with which one can translate C such that it contains p; that is, let Cp WD ft 2 Ed j p 2 t C Cg. Now, it is easy to see that Cp is a convex set of Ed for all p 2 F moreover, F t C C if and only if t 2 \p2F Cp . Thus, Helly’s theorem [65] applied to the convex sets fCp j p 2 Fg implies that F t C C if and only if Cp1 \ Cp2 \ \ Cpd C1 ¤ ; holds for any p1 ; p2 ; : : : ; pd C1 2 F, i.e. if and only if any p1 ; p2 ; : : : ; pd C1 2 F can be covered by a translate of C, finishing the proof of Lemma 6.1.1. Also the following statement plays a central role in our investigations. This is a generalization of the analogue 2-dimensional statement proved in [47]. Lemma 6.1.2. Let F D ff1 ; f2 ; : : : ; fn g be a finite set of points and C be a convex body in Ed ; d 2. Then F cannot be translated into the interior of C if and only if the following two conditions hold. There are closed supporting halfspaces HiC ; HiC ; : : : ; HiC of C assigned to some points of F say, to fi1 ; fi2 ; : : : ; fis with 1 2 s 1 i1 < i2 < < is n and a translation vector t 2 Ed such that (i) The translated point t C fij belongs to the closed halfspace Hij for all 1 j and its boundary s, where the interior of Hij is disjoint from the interior of HiC j C hyperplane is identical to the boundary hyperplane of Hij (which is in fact, a supporting hyperplane of C); (ii) The intersection \sj D1 HiC is nearly bounded, meaning that it lies between two j parallel hyperplanes of Ed . Proof. First, we assume that there are closed supporting halfspaces HiC ; HiC ; 1 2 C : : : ; His of C assigned to some points of F say, to fi1 ; fi2 ; : : : ; fis with 1 i1 < i2 < < is n and a translation vector t 2 Ed satisfying .i / as well as .i i /. Based on this our goal is to show that F cannot be translated into the interior of C or equivalently that F cannot be covered by a translate of the interior intC of C. We prove this in an indirect way: we assume that F can be covered by a translate of intC and look for a contradiction. Indeed, if F can be covered by a translate of intC, then t C F can be covered by a translate of intC; that is, there is a translation vector t 2 Ed such that t C F t C intC. In particular, if F WD ff i1 ;sfi2 ; : :: ; fis g,Cthen t C F t C intC. Clearly, this implies that \sj D1 HiC int \j D1 t C Hij , a j contradiction to .ii /. Second, we assume that F cannot be translated into the interior of C and look for closed supporting halfspaces HiC ; HiC ; : : : ; HiC of C assigned to some points 1 2 s of F say, to fi1 ; fi2 ; : : : ; fis with 1 i1 < i2 < < is n and a translation vector t 2 Ed satisfying .i / as well as .ii /. In order to simplify matters let us start to investigate the case when C is a smooth convex body in Ed , that is, when through each boundary point of C there exists precisely one supporting hyperplane of C. (Also, without loss of generality we assume that the origin o of Ed is an interior point of C.) As F cannot be translated into intC therefore Lemma 6.1.1 implies that there are m d C 1 points of F say, Fm WD ffj1 ; fj2 ; : : : ; fjm g

6.1 Proof of Theorem 5.2.1

101

with 1 j1 < j2 < < jm n such that Fm cannot be translated into intC. Now, let 0 WD inff > 0 j Fm cannot be translated into intCg. Clearly, 0 1 and 0 Fm cannot be translated into intC; moreover, as 0 D supfı > 0 j ıFm can be translated into Cg, therefore there exists a translation vector t 2 Ed such that t C 0 Fm C. Let t C 0 fi1 ; t C 0 fi2 ; : : : ; t C 0 fis with 1 i1 < i2 < < is n; 2 s m d C 1 denote the points of t C 0 Fm that are boundary points of C and let HiC ; HiC ; : : : ; HiC be the corresponding closed 1 2 s supporting halfspaces of C. We claim that HC WD \skD1 HiC is nearly bounded. k C Indeed, if H were not nearly bounded, then there would be a translation vector t0 2 Ed with HC t0 C intHC . As C is a smooth convex body therefore this would imply the existence of a sufficiently small > 0 with the property that ft C 0 fi1 ; t C 0 fi2 ; : : : ; t C 0 fis g t0 C intC, a contradiction. Thus, as o 2 intC therefore the points fi1 ; fi2 ; : : : ; fis and the closed supporting halfspaces HiC ; HiC ; : : : ; HiC and the translation vector t 2 Ed satisfy .i / as well as .ii /. 1 2 s We are left with the case when C is not necessarily a smooth convex body in Ed . In this case let CN ; N D 1; 2; : : : be a sequence of smooth convex bodies lying in intC with limN !C1 CN D C. As F cannot be translated into the interior of CN for all N D 1; 2; : : : therefore applying the method described above to each CN and taking proper subsequences if necessary we end up with some points of F say, fi1 ; fi2 ; : : : ; fis with 1 i1 < i2 < < is n and with s convergent C C C sequences of closed supporting halfspaces HN;i ; HN;i ; : : : ; HN;i of CN and a 1 2 s convergent sequence of translation vectors tN that satisfy .i / and .i i / for each N . By C C taking 0 the limits HiC WD limN !C1 HN;i ; HiC WD limN !C1 HN;i ; : : : ; HiC WD 1 2 s 1 2 C limN !C1 HN;is , and t WD limN !C1 tN we get the desired nearly bounded family of closed supporting halfspaces of C and the translation vector t 2 Ed satisfying .i / as well as .ii /. This completes the proof of Lemma 6.1.2.

6.1.2 From Generalized Billiard Trajectories to Shortest Ones Lemma 6.1.3. Let C be a convex body in Ed ; d 2. If P is a generalized billiard trajectory in C, then P cannot be translated into the interior of C. Proof. Let p1 ; p2 ; : : : ; pn be the vertices of P and let v1 ; v2 ; : : : ; vn be the points of the unit sphere Sd 1 centered at the origin o in Ed whose position vectors are parallel to the inner angle bisectors (halflines) of P at the vertices p1 ; p2 ; : : : ; pn of P. Moreover, let H1C ; H2C ; : : : ; HnC denote the closed supporting halfspaces of C whose boundary hyperplanes are perpendicular to the inner angle bisectors of P at the vertices p1 ; p2 ; : : : ; pn . Based on Lemma 6.1.2 in order to prove that P cannot be translated into the interior of C it is sufficient to show that \niD1 HiC is nearly bounded or equivalently that o 2 conv.fv1 ; v2 ; : : : ; vn g/, where conv.:/ denotes the convex hull of the corresponding set in Ed . It is easy to see that o 2 conv.fv1 ; v2 ; : : : ; vn g/ if and only if for any hyperplane H of Ed passing through o

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and for any of the two closed halfspaces bounded by H say, for H C , we have that H C \ conv.fv1 ; v2 ; : : : ; vn g/ ¤ ;. Indeed, for a given H C let t 2 Ed be chosen so that t C H C is a supporting halfspace of conv.fp1 ; p2 ; : : : ; pn g/. Clearly, at least one vertex of P say, pi0 must belong to the boundary of t C H C and therefore vi0 2 H C \ conv.fv1 ; v2 ; : : : ; vn g/, finishing the proof of Lemma 6.1.3. For the purpose of the following statement it seems natural to introduce generalized .d C 1/-gons in Ed as closed polygonal paths (possibly with selfintersections) having at most d C 1 sides. Theorem 6.1.4. Let C be a convex body in Ed ; d 2 and let Fd C1 .C/ denote the family of all generalized .d C 1/-gons of Ed that cannot be translated into the interior of C. Then Fd C1 .C/ possesses a minimal length member; moreover, the shortest perimeter members of Fd C1 .C/ are identical (up to translations) with the shortest generalized billiard trajectories of C. Proof. If P is an arbitrary generalized billiard trajectory of the convex body C in Ed with vertices p1 ; p2 ; : : : ; pn , then according to Lemma 6.1.3 P cannot be translated into the interior of C. Thus, by Lemma 6.1.1 P possesses at most d C 1 vertices say, pi1 ; pi2 ; : : : ; pid C1 with 1 i1 i2 id C1 n such that pi1 ; pi2 ; : : : ; pid C1 cannot be translated into the interior of C. This implies that by connecting the consecutive points of pi1 ; pi2 ; : : : ; pid C1 by line segments according to their cyclic ordering the generalized .d C 1/-gon Pd C1 obtained, has length l.Pd C1 / at most as large as the length l.P/ of P; moreover, Pd C1 cannot be covered by a translate of intC (i.e., Pd C1 2 Fd C1 .C/). Now, by looking at only those members of Fd C1 .C/ that lie in a d -dimensional ball of sufficiently large radius in Ed we get via a standard compactness argument and Lemma 6.1.2 that Fd C1 .C/ possesses a member of minimal length say, d C1 .C/. As the inequalities l.d C1 .C// l.Pd C1 / l.P/ hold for any generalized billiard trajectory P of C, therefore in order to finish our proof it is sufficient to show that d C1 .C/ is a generalized billiard trajectory of C. Indeed, as d C1 .C/ 2 Fd C1 .C/ therefore d C1 .C/ cannot be translated into intC. Thus, the minimality of d C1 .C/ and Lemma 6.1.2 imply that if q1 ; q2 ; : : : ; qm denote the vertices of d C1 .C/ with m d C1, then there are closed supporting halfspaces H1C ; H2C ; : : : ; HmC of C whose boundary hyperplanes H1 ; H2 ; : : : ; Hm pass through the points q1 ; q2 ; : : : ; qm (each being a boundary C d point of C) and have the property that \m i D1 Hi is nearly bounded in E . If the inner angle bisector at a vertex of d C1 .C/ say, at qi were not perpendicular to Hi , then it is easy to see via Lemma 6.1.2 that one could slightly move qi along Hi to a new position q0i (which is typically an exterior point of C on Hi ) such that the new generalized .d C 1/-gon 0d C1 .C/ 2 Fd C1 .C/ would have a shorter length, a contradiction. This completes the proof of Lemma 6.1.4. Finally, notice that Theorem 5.2.1 follows from Theorem 6.1.4 in a straightforward way.

6.2 Proof of Theorem 5.4.1

103

6.2 Proof of Theorem 5.4.1 Theorem 5.4.1. If P and P0 are two combinatorially equivalent standard ball-polyhedra with equal corresponding face angles in E3 , then P and P0 have equal corresponding inner dihedral angles. We follow the approach of [52]. First, we need to recall the two main ideas of the original proof of Cauchy’s rigidity theorem [73]. The following is called the (spherical) Legendre-Cauchy lemma (see Theorem 22.2 and the discussions followed in [156] as well as [163] for a recent proof and the history of the statement). Lemma 6.2.1. Let U and U 0 be two spherically convex polygons (on an open hemisphere) of the unit sphere S2 WD fy 2 E3 j ko yk D 1g with vertices u1 ; u2 ; : : : ; un , and u01 ; u02 ; : : : ; u0n (enumerated in some cyclic order) and with equal corresponding spherical side lengths (or, equivalently, with kui C1 ui k D ku0i C1 u0i k for all 1 i n, where unC1 WD u1 and u0nC1 WD u01 ). If i and i0 are the angular measures of the interior angles †ui 1 ui ui C1 and †u0i 1 u0i u0i C1 of U and U 0 at the vertices ui and u0i for 1 i n, then either there are at least four sign changes in the cyclic sequence 1 10 ; 2 20 ; : : : ; n n0 (in which we simply ignore the zeros) or the cyclic sequence consists of zeros only. The following is called the sign counting lemma (see Lemma 26.5 in [156] as well as the Proposition in Chap. 10 of [1]). For the purpose of that statement we recall here that a graph is a pair G WD .V; E/, where V is the set of vertices, E is the set of edges, and each edge e 2 E “connects” two vertices v; w 2 V . The graph is called simple if it has no loops (i.e., edges for which both ends coincides) or parallel edges (that have the same set of end vertices). In particular, a graph is planar if it can be drawn on S2 (or, equivalently, in E2 ) without crossing edges. We talk of a plane graph if such a drawing is already given and fixed. Lemma 6.2.2. Suppose that the edges of a simple plane graph are labeled with 0, C and such that around each vertex either all labels are 0 or there are at least four sign changes (in the cyclic order of the edges around the vertex). Then all signs are 0. Second, let P and P0 be two combinatorially equivalent standard ball-polyhedra with equal corresponding face angles in E3 . Clearly, their edge-graph G is a simple plane graph. Now, we assign a label C, 0 or to each edge of G according to whether the inner dihedral angle of the corresponding edge of P is greater than, equal to, or less than the inner dihedral angle of the corresponding edge of P0 . Next, recall that the vertex figure of a vertex v (resp., v0 ) of P (resp., P0 ) is a spherically convex polygon (on an open hemisphere) of the unit sphere S2 with side lengths equal to the face angles meeting at the vertex v (resp., v0 ), and with angles equal to the inner dihedral angles of the edges of P (resp., P0 ) meeting at the vertex v (resp., v0 ). (For more details we refer the interested reader to (6.2) and the relevant

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discussion.) Thus, Lemma 6.2.1 implies that around each vertex of the labelled graph G either all labels are 0 or there are at least four sign changes (in the cyclic order of the edges around the vertex). Finally, Lemma 6.2.2 applied to the labelled graph G implies that all signs must be 0, finishing the proof of Theorem 5.4.1.

6.3 Proof of Theorem 5.4.2 Theorem 5.4.2. If P and P0 are two combinatorially equivalent standard ballpolyhedra with equal corresponding edge lengths and inner dihedral angles in E3 , then P and P0 are congruent. We follow the ideas of the original proof of Stoker’s theorem [172] (see also the proof of Theorem 26.9 in [156]) with properly adjusting that to the family of standard ball-polyhedra. The details are as follows. We need to introduce some basic notation and make some simple observations. In what follows x stands for the notation of a point as well as of its position vector in E3 with o denoting the origin of E3 . Moreover, h; i denotes the standard inner productp in E3 and so, the corresponding standard norm is labelled by k k satisfying kxk D hx; xi. The closed ball of unit radius (or simply the unit ball) centered at x is denoted by BŒx WD fy 2 E3 j kx yk 1g and its boundary bd.BŒx/ WD fy 2 E3 j kx yk D 1g, the unit sphere with center x, is labelled by S.x/ WD bd.BŒx/. f Let P WD \kD1 BŒxk be a standard ball-polyhedron generated by the reduced family fBŒxk j 1 k f g of f 4 unit balls. Here, each unit ball BŒxk gives rise to a face of P namely, to Fk WD S.xk / \ bd.P/ for 1 k f . Clearly, as P is a standard ball-polyhedron, each edge of P is of the form Fk1 \ Fk2 for properly chosen 1 k1 ; k2 f and therefore it can be labelled accordingly with Efk1 ;k2 g . Furthermore, let fEfi;kg j i 2 Ik f1; 2; : : : ; f gg be the family of the edges of Fk . Moreover, let fvj j 1 j vg denote the vertices of P. In particular, let the set of the vertices of Fk be fvj j j 2 Jk f1; 2; : : : ; vgg. Next, let ˛fk1 ;k2 g (resp., ˇj;k ) denote the inner dihedral angle along the edge Efk1 ;k2 g of P (resp., the face angle at the vertex vj of the face Fk of P). Finally, let Ck Œz; WD fy 2 S.xk / j hz xk ; y xk i cos g denote the closed spherical cap lying on S.xk / and having angular radius 0 < with center z 2 S.xk /. Then it is rather easy to show that Fk D

\ i 2Ik

h ˛fi;kg i ; Ck zi;k ; 2 ˛

(6.1)

1 where zi;k WD xk C kxi x .xi xk /. As fi;kg < 2 therefore (6.1) implies that 2 kk Fk is a spherically convex subset of S.xk / (meaning that with any two points of Fk the geodesic arc of S.xk / connecting them lies in Fk ). Furthermore, (6.1) yields that the edges fEfi;kg j i 2 Ik g of Fk are circular arcs of Euclidean radii ˛ fsin fi;kg j i 2 Ik g. Now, let the tangent cone Tvj of P at the vertex vj be 2

6.3 Proof of Theorem 5.4.2

105

defined by Tvj WD cl vj C posfy vj j y 2 Pg , where cl./ (resp., posfg) stands for the closure (resp., positive hull) of the corresponding set. Then it is natural to define the (outer) normal cone Tvj of P at the vertex vj via Tvj WD vj C fy 2 E3 j hy vj ; z vj i 0 for all z 2 Tvj g. Clearly, Tvj as well as Tvj are convex cones of E3 with vj as a common apex. Based on this, it is immediate to define the vertex figure Tvj WD Tvj \ S.vj / as well as the normal image Tvj WD Tvj \ S.vj / of P at the vertex vj . Now, it is straightforward to make the following two observations. The vertex figure Tvj of P at vj is a spherically convex polygon of S.vj / with side lengths (resp., angles) equal to fˇj;k j vj 2 Fk g .resp:; f˛fk1 ;k2 g j vj 2 Efk1 ;k2 g g/

(6.2)

The normal image Tvj of P at vj is a spherically convex polygon of S.vj / with side lengths (resp., angles) equal to f ˛fk1 ;k2 g j vj 2 Efk1 ;k2 g g .resp:; f ˇj;k j vj 2 Fk g/

(6.3)

Having discussed all this, we are ready to take the standard ball-polyhedron P0 WD f \kD1 BŒx0k that is combinatorially equivalent to P. The analogues of the above introduced notations for P0 are as follows: fFk0 WD S.x0k / \ bd.P0 / j 1 k f g; 0 0 0 fEfi;kg j i 2 Ik g; fv0j j 1 j vg; fv0j j j 2 Jk g; ˛fk ; ˇj;k ; Tv0j ; Tv0 ; and 1 ;k2 g j

Ck0 Œz0 ; WD fy0 2 S.x0k / j hz0 x0k ; y0 x0k i cos g with z0 2 S.x0k /; 0 < . 0 By assumption, P and P0 have equal inner dihedral angles, i.e., ˛fk1 ;k2 g D ˛fk . 1 ;k2 g Thus, the analogue of (6.1) reads as follows: h \ ˛fi;kg i ; (6.4) Fk0 D Ck0 z0i;k ; 2 i 2I k

where z0i;k WD x0k C

1 .x0 kx0i x0k k i

x0k /. In particular, the normal image Tv0 of P0 at v0j j

is a spherically convex polygon of S.v0j / with side lengths (resp., angles) equal to 0 j vj 2 Fk g/ f ˛fk1 ;k2 g j vj 2 Efk1 ;k2 g g .resp:; f ˇj;k

(6.5)

Now, we are set for the final approach in proving Theorem 5.4.2. By assumption P and P0 are two combinatorially equivalent standard ball-polyhedra with equal 0 corresponding edge lengths and inner dihedral angles in E3 . Thus, ˛fk1 ;k2 g D ˛fk 1 ;k2 g and (6.1) implies that the corresponding edges of the families fEfi;kg j i 2 Ik g and 0 fEfi;kg j i 2 Ik g of the edges of Fk and Fk0 are circular arcs of equal Euclidean ˛ radii (namely, sin fi;kg 2 ) and of equal length (with the latter property holding by assumption). Hence, in order to complete the proof of Theorem 5.4.2 it is sufficient 0 to show that the corresponding face angles of P and P0 are equal, i.e., ˇj;k D ˇj;k . 0 So, let us compare those face angles by taking ˇj;k ˇj;k . Now, applying the Legendre-Cauchy lemma (i.e., Lemma 6.2.1) to the normal images Tvj and Tv0 and j

using (6.3) as well as (6.5) we obtain the following result.

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Sublemma 6.3.1 Let vj ; 1 j v be an arbitrary vertex of the standard ball-polyhedron P. Then either there are at least four sign changes in the cyclic 0 sequence of the face angle differences fˇj;k ˇj;k j vj 2 Fk g around the vertex vj of P or the cyclic sequence in question consists of zeros only. According to (6.1) (resp., (6.4)) Fk (resp., Fk0 ) is a spherically convex subset of the unit sphere S.xk / (resp., S.x0k /) for any 1 k f and therefore the spherical 0 convex hull F k (resp., F k ) of the vertices fvj j j 2 Jk g (resp., fv0j j j 2 Jk g) of Fk (resp., Fk0 ) on S.xk / (resp., S.x0k )) clearly possesses the property that F k Fk 0 0 (resp., F k Fk0 ). Moreover, if ˇ j;k (resp., ˇ j;k ) denotes the angular measure of 0

the interior angle of F k (resp., F k ) at the vertex vj (resp., v0j ), then (6.1) and (6.4) imply again in a straightforward way that the corresponding side lengths of F k and 0 0 0 F k are equal furthermore, ˇj;k ˇj;k D ˇ j;k ˇ j;k holds for any vertex vj ; j 2 Jk 0

of Fk . Thus, Lemma 6.2.1 applied to F k and F k proves the following statement. Sublemma 6.3.2 Let Fk ; 1 k f be an arbitrary face of the standard ball-polyhedron P. Then either there are at least four sign changes in the cyclic 0 sequence of the face angle differences fˇj;k ˇj;k j vj 2 Fk g around the face Fk of P or the cyclic sequence in question consists of zeros only. Finally, let us take the medial graph G of P with “vertices” corresponding to the edges of P and with “edges” connecting two “vertices” if the corresponding two edges of P are adjacent (i.e., share a vertex in common) and lie on the same face of P. So, if the “edge” of G “connects” the two edges of P that lie on the face Fk of P and have the vertex vj in common enclosing the face angle ˇj;k , then we label the 0 “edge” in question of G by sign.ˇj;k ˇj;k /, where sign.ı/ is C; or 0 depending on whether ı is positive, negative or zero. Thus, using Sublemmas 6.3.1 and 6.3.2, one can apply the sign counting lemma (i.e., Lemma 6.2.2) to the dual graph G of 0 G concluding in a straightforward way that ˇj;k ˇj;k D 0. This finishes the proof of Theorem 5.4.2.

6.4 Underlying Truncated Delaunay Complex of a Ball-Polyhedron In this section we introduce some basic notions and tools that are needed for our proof of Theorems 5.4.6 and 5.4.10. Definition 6.4.1. A convex polyhedron of E3 is a bounded intersection of finitely many closed half-spaces in E3 . A polyhedral complex in E3 is a finite family of convex polyhedra such that any vertex, edge, and face of a member of the family is again a member of the family, and the intersection of any two members is empty or a vertex or an edge or a face of both members.

6.4 Underlying Truncated Delaunay Complex of a Ball-Polyhedron

107

Second, let us give a detailed construction of the so-called truncated Delaunay complex of a ball-polyhedron. We leave the proofs of the claims mentioned here to the reader partly because they are straightforward and partly because they are also well known. (For more details we refer the interested reader to [11, 171], and [95].) The farthest-point Voronoi tiling corresponding to a finite set C WD fc1 ; : : : ; cn g in E3 is the family V WD fV1 ; : : : ; Vn g of closed convex polyhedral sets Vi WD fx 2 E3 W kx ci k kx cj k for all j ¤ i; 1 j ng, 1 i n. (Here a closed convex polyhedral set means a not necessarily bounded intersection of finitely many closed half-spaces in E3 .) We call the elements of V farthest-point Voronoi cells. In the sequel we omit the words “farthest-point” as we do not use the other (more popular) Voronoi tiling: the one capturing closest points. It is known that V is a tiling of E3 . We call the vertices, (possibly unbounded) edges and (possibly unbounded) faces of the Voronoi cells of V simply the vertices, edges and faces of V. The truncated Voronoi tiling corresponding to C is the family V t of the closed convex sets fV1 \ BŒc1 ; : : : ; Vn \ BŒcn g. Clearly, from the definition it follows that V t D fV1 \ P; : : : ; Vn \ Pg where P WD BŒc1 \ : : : \ BŒcn . We call the elements of V t truncated Voronoi cells. Next, we define the (farthest-point) Delaunay complex D assigned to the finite set C D fc1 ; : : : ; cn g E3 . It is a polyhedral complex on the vertex set C . For an index set I f1; : : : ; ng, the convex polyhedron convfci j i 2 I g is a member of D if and only if there is a point p in \i 2I Vi which is not contained in any other Voronoi cell, where convfg stands for the convex hull of the corresponding set. In other words, convfci j i 2 I g 2 D if and only if there is a point p 2 E3 and a radius > 0 such that fci j i 2 I g bd.B.p; // and fci j i … I g B.p; /, where B.p; / stands for the open ball having radius and center point p in E3 . It is known that D is a polyhedral complex moreover, it is a tiling of convfc1 ; : : : ; cn g by convex polyhedra. The more exact connection between the Voronoi tiling V and the Delaunay complex D is described in the following statement. (In what follows, dim./ refers to the dimension of the given set, i.e., dim./ stands for the dimension of the smallest dimensional affine subspace containing the given set.) Lemma 6.4.2. Let C D fc1 ; : : : ; cn g E3 be a finite set, and V D fV1 ; : : : ; Vn g be the corresponding Voronoi tiling of E3 . (V) For any vertex p of V there exists an index set I f1; : : : ; ng with dim.fci j i 2 I g/ D 3 such that convfci j i 2 I g 2 D and p D \i 2I Vi . Vica versa, if I f1; : : : ; ng with dim.fci j i 2 I g/ D 3 and convfci j i 2 I g 2 D, then \i 2I Vi is a vertex of V. (E) For any edge E of V there exists an index set I f1; : : : ; ng with dim.fci j i 2 I g/ D 2 such that convfci j i 2 I g 2 D and E D \i 2I Vi . Vica versa, if I f1; : : : ; ng with dim.fci j i 2 I g/ D 2 and convfci j i 2 I g 2 D, then \i 2I Vi is an edge of V. (F) For any face F of V there exists an index set I f1; : : : ; ng of cardinality 2 such that convfci j i 2 I g 2 D and F D \i 2I Vi . Vica versa, if I f1; : : : ; ng of cardinality 2 and convfci j i 2 I g 2 D, then \i 2I Vi is a face of V.

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Fig. 6.1 Let us take four points, c1 ; : : : ; c4 in E2 . The bold solid lines bound the four Voronoi cells, V1 ; : : : ; V4 . The bold dashed circular arcs bound the planar ball-polyhedron – a disk-polygon. (We note that for the sake of simplicity, the generating disks of the disk-polygons constructed here are not necessarily of unit radius.) The part of each Voronoi cell inside the disk-polygon is the corresponding truncated Voronoi cell. On the first example, the truncated Delaunay complex coincides with the non-truncated one. On the second example, the Voronoi and the Delaunay complexes are the same as on the first, but the truncated Voronoi and Delaunay complexes are different

Finally, we define the truncated Delaunay complex Dt assigned to C similarly to D. For an index set I f1; : : : ; ng, the convex polyhedron convfci j i 2 I g 2 D is a member of Dt if and only if there is a point p in \i 2I .Vi \ BŒci / which is not contained in any other truncated Voronoi cell. Recall that the truncated Voronoi cells are contained in the ball-polyhedron P D BŒc1 \ : : : \ BŒcn . Thus, convfci j i 2 I g 2 Dt if and only if there exists a point p 2 P and a radius > 0 such that fci j i 2

6.5 Proof of Theorem 5.4.6

109

I g bd.B.p; // and fci j i … I g B.p; /. For the convenience of the reader Fig. 6.1 gives a summary of the concepts of this section in the 2-dimensional case.

6.5 Proof of Theorem 5.4.6 Theorem 5.4.6. Every normal ball-polyhedron of E3 is globally rigid with respect to its inner dihedral angles within the family of normal ball-polyhedra. f

Let P WD \kD1 BŒxk be an arbitrary normal ball-polyhedron of E3 generated by the reduced family fBŒxk j 1 k f g of f 4 unit balls. We note that the condition being reduced implies that the center points fx1 ; : : : ; xf g are in (strictly) convex position in E3 . Let CP WD convfx1 ; : : : ; xf g be the center-polyhedron of P in E3 with the face lattice induced by the Delaunay complex D assigned to the point set fx1 ; : : : ; xf g. (Recall that D is a tiling of CP .) The following is the core part of our proof of Theorem 5.4.6. Lemma 6.5.1. Any normal ball-polyhedron P of E3 is a standard ball-polyhedron with its face lattice being dual to the face lattice of its center-polyhedron CP . Proof. First, let us take an arbitrary circumscribed sphere say, S.x; ı/ of the f flower-polyhedron P[ D [kD1 BŒxk having center point x and radius ı. By definition there exists at least one such S.x; ı/ moreover, by assumption 0 < ı < 2. Let I f1; : : : ; f g denote the set of the indices of the unit balls fBŒxk j 1 k f g that are tangent to S.x; ı/ (with the remaining unit balls lying inside the circumscribed sphere S.x; ı/). Also, let V and D (resp., V t and Dt ) denote the Voronoi tiling and the Delaunay complex (resp., the truncated Voronoi tiling and the truncated Delaunay complex) assigned to the finite set fx1 ; : : : ; xf g. It follows from the definition of S.x; ı/ in a straightforward way that dim.fxi j i 2 I g/ D 3 and convfxi j i 2 I g 2 D. Thus, part .V / of Lemma 6.4.2 clearly implies that x D \i 2I Vi is a vertex of V. Furthermore, 0 < ı < 2 yields that x 2 int.P/ and therefore x is a vertex of V t as well and convfxi j i 2 I g 2 Dt , where int./ stands for the interior of the corresponding set. Second, it is easy to see via part .V / of Lemma 6.4.2 that each vertex x of V is in fact, a center of some circumscribed sphere of the flower-polyhedron P[ . Thus, we obtain that the vertex sets of V and V t are identical (lying in int.P/) and therefore the polyhedral complexes D and Dt are the same, i.e., D Dt . Finally, based on this and using Lemma 6.4.2 again, we get that the vertex-edge-face structure of the normal ball-polyhedron P is dual to the face lattice of the center-polyhedron CP D convfx1 ; : : : ; xf g induced by the polyhedral complex D Dt . This completes the proof of Lemma 6.5.1. Now, let P D \kD1 BŒxk and P0 D \kD1 BŒx0k be two combinatorially equivalent normal ball-polyhedra with equal corresponding inner dihedral angles in E3 . Our goal is to show that P is congruent to P0 . f

f

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Let CP WD convfx1 ; : : : ; xf g (resp., CP0 WD convfx01 ; : : : ; x0f g) be the center-polyhedron of P (resp., P0 ) in E3 with the face lattice induced by the underlying Delaunay complex D (resp., D0 ). By Lemma 6.5.1 each edge of CP (resp., CP0 ) corresponds to an edge of P (resp., P0 ) furthermore, the length of an edge of CP (resp., CP0 ) is determined by the inner dihedral angle of the corresponding edge of P (resp., P0 ). Thus, Lemma 6.5.1 implies that the face lattices of CP and CP0 are isomorphic moreover, the corresponding edges of CP and CP0 are of equal length. As each face of CP (resp., CP0 ) is a convex polygon inscribed in a circle, the corresponding faces of CP and CP0 are congruent. Hence, bd.CP / (resp., bd.CP0 /) are convex polyhedral surfaces in E3 , which are combinatorially equivalent with the corresponding faces being congruent. Thus, by the Cauchy–Alexandrov theorem for polyhedral surfaces (see Theorem 27.6 in [156]) CP is congruent to CP0 and therefore P is congruent to P0 , finishing the proof of Theorem 5.4.6.

6.6 Infinitesimally Rigid Polyhedron and Dual Ball-Polyhedron In this section we introduce some additional notions and tools that are needed for our proof of Theorem 5.4.10. Definition 6.6.1. A polyhedron of E3 is simply the union of all members of a 3dimensional polyhedral complex in E3 possessing the additional property that its (topological) boundary in E3 is a surface in E3 (i.e., a 2-dimensional topological manifold embedded in E3 ). Definition 6.6.2. A polyhedron Q of E3 is • Weakly convex if its vertices are in convex position (i.e., if its vertices are the vertices of a convex polyhedron); • Co-decomposable if its complement in conv.Q/ can be triangulated (i.e., obtained as a simplicial complex) without adding new vertices; Q such • Weakly co-decomposable if it is contained in a convex polyhedron Q, Q and the complement of Q in Q Q can be that all vertices of Q are vertices of Q, triangulated without adding new vertices. Next, we give a very brief description of a collection of notions from the theory of rigidity of tensegrity frameworks. For more details on this as well as for an overview on the relevant theory we refer the interested reader to [79]. The short details are as follows. The boundary of every polyhedron in E3 is the disjoint union of planar convex polygons, and hence it can be triangulated without adding new vertices. Now, let P be a polyhedron in E3 and let T be a triangulation of its boundary without adding new vertices. We call the 1-skeleton G.T / of T the edge graph of T . By an infinitesimal flex of the edge graph G.T / in E3 we mean an assignment of vectors to the vertices of G.T / (i.e., to the vertices of P) such that the displacements of

6.7 Proof of Theorem 5.4.10

111

the vertices in the assigned directions induce a zero first-order change of the edge lengths: hpi pj ; qi qj i D 0 for every edge pi pj of G.T /, where qi is the vector assigned to the vertex pi . An infinitesimal flex is called trivial if it is the restriction of an infinitesimal rigid motion of E3 . All this leads us to the following fundamental concept. Definition 6.6.3. We say that the polyhedron P E3 is infinitesimally rigid if every infinitesimal flex of the edge graph G.T / of a triangulation T of the boundary of P is trivial. It is not hard to see that the infinitesimal rigidity of a polyhedron is a well-defined notion, i.e., independent of the triangulation T . We need the following remarkable rigidity theorem of Izmestiev and Schlenker [132] for our proof of Theorem 5.4.10. Theorem 6.6.4. Every weakly co-decomposable polyhedron of E3 is infinitesimally rigid. It is convenient to have the following notation as well. For a set C E3 we denote the intersection of closed unit balls with centers in C by BŒC WD \fBŒcW c 2 C g. Recall that every ball-polyhedron P D BŒC can be generated such that BŒC n fcg ¤ BŒC holds for any c 2 C . Therefore whenever we take a ball-polyhedron P D BŒC we always assume the above mentioned reduced property of C . The following duality theorem has been proved in [52] and it is also needed for our proof of Theorem 5.4.10. Theorem 6.6.5. Let P be a standard ball-polyhedron of E3 . Then the intersection P of the closed unit balls centered at the vertices of P is another standard ballpolyhedron whose face lattice is dual to that of P (i.e., there exists an order reversing bijection between the face lattices of P and P ). For a more recent discussion on the above duality theorem and its generalization we refer the interested reader to [142].

6.7 Proof of Theorem 5.4.10 Theorem 5.4.10. Let P be a simple and standard ball-polyhedron of E3 . Then P is locally rigid with respect to its inner dihedral angles. Our proof is based on the following three lemmas. Lemma 6.7.1. Let P D BŒC be a simple ball-polyhedron in E3 . Then no vertex of the Voronoi tiling V corresponding to C is on bd.P/, and no edge of V is tangent to P. Proof. By Lemma 6.4.2 (V), at least four Voronoi cells meet in any vertex of V. Moreover, the intersection of each Voronoi cell with bd.P/ is a face of P, since P is

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generated by a reduced set of centers. Hence, if a vertex of V were on bd.P/, then at least four faces of P would meet at a point, contradicting the assumption that P is simple. Let E be an edge of V, and assume that it contains a point p 2 bd.P/. By the previous paragraph, p 2 relint.E/, where relint./ denotes the relative interior of the corresponding set. From Lemma 6.4.2 (E) it follows that p is in the intersection of some Voronoi cells fVi W i 2 I g with dim.fci W i 2 I g/ D 2. Clearly, E is orthogonal to the plane afffci W i 2 I g spanned by the points fci W i 2 I g. Finally, in a neighborhood of p, P is the same as BŒfci W i 2 I g and hence, E must intersect int.P/. Lemma 6.7.2. Let P D BŒC be a simple ball-polyhedron in E3 . Then Dt is a subpolyhedral complex of D, that is Dt D, and faces, edges, and vertices of members of Dt are again members of Dt . Proof. Clearly, Dt D, and their vertex sets are identical (both are C ). Let convfci W i 2 I g 2 Dt be a 3-dimensional member of Dt . Then, the corresponding vertex (Lemma 6.4.2 (V)) v of V is in int.P/ by Lemma 6.7.1. For a given face of convfci W i 2 I g, there is a corresponding edge (Lemma 6.4.2 (E)) E of V. Clearly, v is an endpoint of E. Now, relint.E/ \ P ¤ ;, and thus the face convfci W i 2 I g of V corresponding to E is in Dt . Next, let convfci W i 2 I g 2 Dt be a 2-dimensional member of Dt . Then, for the corresponding edge E of V we have relint.E/ \ P ¤ ;. By Lemma 6.7.1, E is not tangent to P, thus relint.E/ \ int.P/ ¤ ;. An edge ci cj of convfci W i 2 I g corresponds to a face (Lemma 6.4.2 (F)) F of V. Clearly, E is an edge of F . Now, relint.F / \ P ¤ ;, and thus ci cj is in Dt . The following lemma helps to understand the 2-dimensional members of Dt . Let P D BŒC be a simple and standard ball-polyhedron in E3 . Denote by Q the polyhedral complex formed by the 3-dimensional members of Dt and all of their faces, edges and vertices (i.e., we drop “hanging” faces/edges/vertices of Dt , that is, those faces/edges/vertices that do not belong to a 3-dimensional member). Clearly, [Q is a subset of E3 and thus, its boundary is defined. We equip this boundary with a polyhedral complex structure in the obvious way as follows: we define the boundary of Q as the collection of those faces, edges and vertices of Q that lie on the boundary of [Q. We denote this polyhedral complex by bd Q. Lemma 6.7.3. Let P D BŒC be a simple and standard ball-polyhedron in E3 and Q be defined as above. Then the 2-dimensional members of bd.Q/ are triangles, and a triangle convfc1 ; c2 ; c3 g is in bd.Q/ if, and only if, the corresponding faces F1 ; F2 ; F3 of P meet (at a vertex of P). Proof. By Lemma 6.7.2, the 2-dimensional members of bd.Q/ are 2-dimensional members of Dt . Let convfci W i 2 I g 2 Dt with dim.fci W i 2 I g/ D 2. Then, clearly, convfci W i 2 I g 2 D and, by Lemma 6.4.2 (E), it corresponds to an edge E of V which intersects P. Now, E is a closed line segment, or a closed ray, or a line. By Lemma 6.7.1, E is not tangent to P, and (by Lemma 6.7.1) E has no

6.7 Proof of Theorem 5.4.10

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endpoint on bd.P/. Thus, E intersects the interior of P. We claim that E has at least one endpoint in int.P/. Suppose, it does not. Then E \ bd.P/ is a pair of points and so, the faces of P corresponding to indices in I meet at more than one point. Since jI j 3, it contradicts the assumption that P is standard. We remark that this is a crucial point where we used the standardness of P. So, E has either one or two endpoints in int.P/. If it has two, then the two distinct 3-dimensional Delaunay cells corresponding to those endpoints (as in Lemma 6.4.2 (V)) are both members of Dt and contain the planar convex polygon convfcI W i 2 I g, and thus, convfcI W i 2 I g is not on the boundary of Q. If E has one endpoint in int.P/, then there is a unique 3-dimensional polyhedron in Dt (the one corresponding to that endpoint of E) that contains the planar convex polygon convfci W i 2 I g. Moreover, in this case E intersects bd.P/ at a vertex of P. Since P is simple, that vertex is contained in exactly three faces of P, and hence, convfci W i 2 I g is a triangle. Next, working in the reverse direction, assume that F1 ; F2 ; F3 are faces of P that meet at a vertex v of P. Then v is in exactly three Voronoi cells, V1 ; V2 and V3 . Thus, convfc1 ; c2 ; c3 g 2 D, and E WD V1 \ V2 \ V3 is an edge of V. By the above argument, E has one endpoint in P and so, convfc1 ; c2 ; c3 g is a member of Dt , and has the property that exactly one 3-dimensional member of Dt contains it. If follows that convfc1 ; c2 ; c3 g is in bd.Q/. From the last paragraph of the proof and the fact that P has at least one vertex, we can deduce the following Remark 6.7.4. With the notations and the assumptions of Lemma 6.7.3, Dt contains at least one 3-dimensional cell, and the vertex set of Q is C . We recall the following important notion. Definition 6.7.5. The nerve of a family G of sets is the abstract simplicial complex N .G/ WD ffGi W i 2 I gW Gi 2 G for all i 2 I and \ Gi ¤ ;g. i 2I

Now, let P D BŒC be a simple and standard ball-polyhedron in E3 and let F denote the set of its faces. Let S be the abstract simplicial complex on the vertex set C generated by the 2-dimensional members of bd.Q/, which are, according to Lemma 6.7.3, certain triplets of points in C . Both S and the nerve N .F / of F are 2-dimensional abstract simplicial complexes. We claim that they both have the following “edge property”: any edge is contained in a 2-dimensional simplex. Indeed, S has this property by definition, since it is a simplicial complex generated by a family of 2-dimensional simplices. On the other hand, N .F / also has this property, because P is simple and standard, and hence any edge of P has a vertex as an endpoint which is a point of intersection of three faces of P. Consider the mapping W ci 7! Fi that maps each center point in C to the corresponding face of P. This is a bijection between the 0-dimensional members of S and the 0-dimensional members of N .F /. By Lemma 6.7.3 the 2-dimensional members of S correspond via to the 2-dimensional members of N .F /. By the “edge property” in the previous paragraph, it follows that is an isomorphism of the two abstract simplicial complexes, S and N .F /.

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By Theorem 6.6.5, N .F / is isomorphic to the face-lattice of another standard ball-polyhedron: P . Since P is a convex body in E3 (i.e., a compact convex set with non-empty interior in E3 ), the union of its faces is homeomorphic to the 2-sphere. Thus, S as an abstract simplicial complex is homeomorphic to the 2sphere. On the other hand, bd.Q/ is a geometric realization of S. Thus, we have obtained that bd.Q/ is a geometric simplicial complex which is homeomorphic to the 2-sphere. It follows that Q is homeomorphic to the 3-ball. So, we have that Q is a polyhedron (the point being: it is topologically nice, that is, its boundary is a surface in E3 ). Clearly, Q is a weakly convex polyhedron as C is in convex position. Furthermore, Q is (co-decomposable as well as) weakly co-decomposable, as Dt is a sub-polyhedral complex of D (by Lemma 6.7.2), which is a polyhedral complex of convex polyhedra whose union is conv.Q/ D conv.C /. So far, we have proved that Q is a weakly co-decomposable polyhedron with triangular faces in E3 . By Theorem 6.6.4, Q is infinitesimally rigid. Since bd.Q/ itself is a geometric simplicial complex therefore its edge graph is rigid because infinitesimal rigidity implies rigidity (for more details on that see [79]). Finally, we recall that the edges of the polyhedron Q correspond to the edges of the ballpolyhedron P, and the lengths of the edges of Q determine (via a one-to-one mapping) the corresponding inner dihedral angles of P. It follows that P is locally rigid with respect to its inner dihedral angles.

6.8 Proofs of Theorems 5.5.1–5.5.3 6.8.1 Strict Separation by Spheres of Radii at Most One Theorem 5.5.1. Let A; B Ed be finite sets. Then A and B can be strictly separated by a sphere S d 1 .c; r/ with r 1 such that A Bd .c; r/ if and only if the following holds. For every T A [ B with card T d C 2, T \ A and T \ B can be strictly separated by a sphere S d 1 .cT ; rT / with rT 1 such that T \ A Bd .cT ; rT /. For the proof of Theorem 5.5.1 we need the following weaker version of it due to Houle [128] as well as the following lemma proved in [59]. Theorem 6.8.1. Let A; B Ed be finite sets. Then A and B can be strictly separated by a sphere S d 1 .c; r/ such that A Bd .c; r/ if and only if for every T A [ B with cardT d C 2, T \ A and T \ B can be strictly separated by a sphere S d 1 .cT ; rT / such that T \ A Bd .cT ; rT /. Lemma 6.8.2. Let A; B Ed be finite sets and suppose that S d 1 .o; 1/ is the smallest sphere that separates A from B such that A Bd Œo; 1. Then there is a set T A [ B with cardT d C 1 such that S d 1 .o; 1/ is the smallest sphere S d 1 .c; r/ that separates T \ A from T \ B and satisfies T \ A Bd Œc; r.

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115

We prove the “if” part of Theorem 5.5.1; the opposite direction is trivial. Theorem 6.8.1 guarantees the existence of the smallest sphere S d 1 .c0 ; r 0 / that separates A and B such that A Bd Œc0 ; r 0 . According to Lemma 6.8.2, there is a set T A [ B with cardT d C 1 such that S d 1 .c0 ; r 0 / is the smallest sphere that separates T \ A from T \ B and whose convex hull contains T \ A. By the assumption, we have r 0 < rT 1. Note that Theorem 6.8.1 guarantees the existence of a sphere S d 1 .c ; r / that strictly separates A from B and satisfies d 1 A Bd .c ; r /. Because r 0 < 1, there is a sphere with r 1 such that d 0S 0 .c; r/ d 0 0 d d d B Œc ; r \ B .c ; r / B .c; r/ E n B .c ; r / [ Bd Œc ; r . This sphere clearly satisfies the conditions in Theorem 5.5.1 and so, the proof of Theorem 5.5.1 is complete.

6.8.2 Characterizing Spindle Convex Sets Theorem 5.5.2. Let A Ed be a compact convex set. Then the following are equivalent. (i) A is spindle convex. (ii) A is the intersection of closed unit balls containing it. (iii) For every boundary point of A, there is a closed unit ball that supports A at that point. Our proof of Theorem 5.5.2 is based on the following statement. Lemma 6.8.3. Let a compact spindle convex set C Ed be supported by the hyperplane H in Ed at x 2 bdC. Then the closed unit ball supported by H at x and lying in the same side as C contains C. Proof. Let Bd Œc; 1 be the closed unit ball that is supported by H at x and is in the same closed half-space bounded by H as C. We show that Bd Œc; 1 is the desired unit ball. Assume that C is not contained in Bd Œc; 1. So, there is a point y 2 C; y … d B Œc; 1. Then, by taking the intersection of the configuration with the plane that contains x; y, and c, we see that there is a shorter unit circular arc connecting x and y that does not intersect Bd .c; 1/. Hence, H cannot be a supporting hyperplane of C at x, a contradiction. Indeed, it is easy to see that Lemma 6.8.3 implies Theorem 5.5.2 in a rather straightforward way.

6.8.3 Separating Spindle Convex Sets Theorem 5.5.3 Let C; D Ed be spindle convex sets. Suppose C and D have disjoint relative interiors. Then there is a closed unit ball Bd Œc; 1 such that C Bd Œc; 1 and D Ed n Bd .c; 1/.

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Furthermore, if C and D have disjoint closures and one, say C, is different from a unit ball, then there is an open unit ball Bd .c; 1/ such that C Bd .c; 1/ and D Ed n Bd Œc; 1. We prove Theorem 5.5.3 as follows. Since C and D are spindle convex, they are convex bounded sets with disjoint relative interiors. So, their closures are convex compact sets with disjoint relative interiors. Hence, they can be separated by a hyperplane H that supports C at a point, say x. The closed unit ball Bd Œc; 1 of Lemma 6.8.3 satisfies the conditions of the first statement of Theorem 5.5.3. For the second statement of Theorem 5.5.3, we assume that C and D have disjoint closures, so Bd Œc; 1 is disjoint from the closure of D and remains so even after a sufficiently small translation. Furthermore, C is a spindle convex set that is different from a unit ball, so c … conv.C \ S d 1 .c; 1//. Hence, there is a sufficiently small translation of Bd Œc; 1 that satisfies the second statement of Theorem 5.5.3, finishing the proof of Theorem 5.5.3.

6.9 Proof of Theorem 5.6.1 Theorem 5.6.1. Let X Ed be a closed set. (i) If y 2 bd.convs X /, then there is a set fx1 ; x2 ; : : : ; xd g X such that y 2 convs fx1 ; x2 ; : : : ; xd g. (ii) If y 2 int.convs X /, then there is a set fx1 ; x2 ; : : : ; xd C1 g X such that y 2 int.convs fx1 ; x2 ; : : : ; xd C1 g/.

6.9.1 Spindle Convex Hulls and Supporting Spheres Let S k .c; r/ Ed be a k-dimensional sphere centered at c and having radius r with 0 k d 1. Recall the following strong version of spherical convexity. A set F S k .c; r/ is spherically convex if it is contained in an open hemisphere of S k .c; r/ and for every x; y 2 F the shorter great-circular arc of S k .c; r/ connecting x with y is in F . The spherical convex hull of a set X S k .c; r/ is defined in the natural way and it exists if and only if X is in an open hemisphere of S k .c; r/. We denote it by Sconv.X; S k .c; r//. Carath´eodory’s theorem can be stated for the sphere in the following way. If X S k .c; r/ is a set in an open hemisphere of S k .c; r/, then Sconv.X; S k .c; r// is the union of spherical simplices with vertices in X . The proof of this spherical equivalent of the original Carath´eodory’s theorem uses the central projection of the open hemisphere of S k .c; r/ to Ek . Recall that the circumradius cr.X / of a bounded set X Ed is defined as the radius of the unique smallest d -dimensional closed ball that contains X (also known

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as the circumball of X ). Now, it is easy to see that if C Ed is a spindle convex set such that C Bd Œq; 1 and cr.C / < 1, then C \ S d 1 .q; 1/ is spherically convex on S d 1 .q; 1/. The following lemma describes the surface of a spindle convex hull. Lemma 6.9.1. Let X Ed be a closed set such that cr.X / < 1 and let Bd Œq; 1 be a closed unit ball containing X . Then (i) X \ S d 1 .q; 1/ is contained in an open hemisphere of S d 1 .q; 1/, (ii) convs .X / \ S d 1 .q; 1/ D Sconv.X \ S d 1 .q; 1/; S d 1 .q; 1//: Proof. Because cr.X / < 1, we obtain that X is contained in the intersection of two distinct closed unit balls which proves (i). Note that by (i), the right-hand side Z WD Sconv.X \ S d 1 .q; 1/; S d 1 .q; 1// of (ii) exists. We show that the set on the left-hand side is contained in Z; the other containment follows from the discussion right before Lemma 6.9.1. Suppose that y 2 convs .X / \ S d 1 .q; 1/ is not contained in Z. We show that there is a hyperplane H through q that strictly separates Z from y. Consider an open hemisphere of S d 1 .q; 1/ that contains Z, call the spherical center of this hemisphere p. If y is an exterior point of the hemisphere, H exists. If y is on the boundary of the hemisphere, then, by moving the hemisphere a little, we find another open hemisphere that contains Z, but with respect to which y is an exterior point. Assume that y is contained in the open hemisphere. Let L be a hyperplane tangent to S d 1 .q; 1/ at p. We project Z and y centrally from q onto L and, by the separation theorem of convex sets in L, we obtain a .d 2/-dimensional affine subspace T of L that strictly separates the image of Z from the image of y. Then H WD aff.T [ fqg/ is the desired hyperplane. Hence, y is contained in one open hemisphere of S d 1 .q; 1/ and Z is in the other. Let v be the unit normal vector of H pointing towards the hemisphere of S d 1 .q; 1/ that contains Z. Since X is closed, its distance from the closed hemisphere containing y is positive. Hence, we can move q a little in the direction v to obtain the point q0 such that X Bd Œq; 1 \ Bd Œq0 ; 1 and y … Bd Œq0 ; 1: As Bd Œq0 ; 1 separates X from y, the latter is not in convs .X /, a contradiction.

6.9.2 Carath´eodory’s Theorem for Spindle Convex Hulls Now, we prove Theorem 5.6.1. Assume that cr.X / > 1. Recall that the intersection of the d -dimensional closed unit balls of Ed centered at the points of X is denoted by BŒX . Then BŒX D ;; hence, by Helly’s theorem, there is a set fx0 ; x1 ; : : : ; xd g X such that BŒfx0 ; x1 ; : : : ; xd g D ;. It follows that convs .fx0 ; x1 ; : : : ; xd g/ D Ed . Thus, (i) and (ii) follow. Now, we prove (i) for cr.X / < 1. By the spherical Carath´eodory theorem, Lemmas 6.8.3, and 6.9.1 we obtain that

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y 2 Sconv.fx1 ; x2 ; : : : ; xd g; S d 1 .q; 1// for some fx1 ; x2 ; : : : ; xd g X and some q 2 Ed such that X Bd Œq; 1. Hence, y 2 convs fx1 ; x2 ; : : : ; xd g. We prove (i) for cr.X / D 1 by a limit argument as follows. Without loss of generality, we may assume that X Bd Œo; 1. Let X k WD .1 k1 /X for any k 2 ZC . Let yk be the point of bd convs .X k / closest to y. Thus, lim y k D y. k!1

Clearly, cr.X k / < 1, hence there is a set fxk1 ; xk2 ; : : : ; xkd g X k such that yk 2 convs fxk1 ; xk2 ; : : : ; xkd g. By compactness, there is a sequence 0 < i1 < i2 < : : : i i i of indices such that all the d sequences fx1j W j 2 ZC g; fx2j W j 2 ZC g; : : : ; fxdj W j 2 ZC g converge. Let their respective limits be x1 ; x2 ; : : : ; xd . Since X is closed, these d points are contained in X . Clearly, y 2 convs fx1 ; x2 ; : : : ; xd g. To prove (ii) for cr.X / 1, suppose that y 2 int .convs X /. Then let x0 2 X \ bd .convs X / be arbitrary and let y1 be the intersection of bd .convs X / with the ray starting from x0 and passing through y. Now, by (i), y1 2 convs fx1 ; x2 ; : : : ; xd g for some fx1 ; x2 ; : : : ; xd g X . Then clearly y 2 int .convs fx0 ; x1 ; : : : ; xd g/.

6.10 Proof of Theorem 5.7.3 Theorem 5.7.3. Let X Ed , d 3 be an arbitrary compact set with diam.X / 1 and let BŒX be the intersection of the closed d -dimensional unit balls centered at the points of X . Then I.BŒX / < 4

12 3

d2 d2 3 3 3 d .3 C ln d / < 5d 2 .4 C ln d / : 2 2 3 2

6.10.1 On the Boundary of Spindle Convex Hulls Let X Ed ; d 3 be a compact set of Euclidean diameter diam.X / 1. Recall that BŒX Ed denotes the convex body which is the intersection of the closed unit balls of Ed centered at the points of X . For the following investigations it is more proper to use the normal images than the Gauss images of the boundary points of BŒX defined as follows. The normal image NBŒX .b/ of the boundary point b 2 bd .BŒX / of BŒX is NBŒX .b/ WD .fbg/

6.10 Proof of Theorem 5.7.3

119

In other words, NBŒX .b/ Sd 1 is the set of inward unit normal vectors of all hyperplanes that support BŒX at b. Clearly, NBŒX .b/ is a closed spherically convex subset of Sd 1 . (Here we refer to the strong version of spherical convexity introduced for Lemma 6.9.1.) We need to introduce the following notation as follows. For a set A Sd 1 let C A D fx 2 Sd 1 j hx; yi > 0 for all y 2 Ag. (Here k k and h; i refer to the canonical Euclidean norm and the canonical inner product on Ed .) As is well known, illumination can be reformulated as follows: The direction u 2 Sd 1 illuminates the boundary point b of the convex body BŒX if and only if u 2 NBŒX .b/C . (Because the proof of this claim is straightforward we leave it to the reader. For more insight on illumination we refer the interested reader to [28] and the relevant references listed there.) Finally, we need to recall some further notations as well. Let a and b be two points in Ed . If ka bk < 2, then the (closed) spindle of a and b, denoted by Œa; bs , is defined as the union of circular arcs with endpoints a and b that are of radii at least one and are shorter than a semicircle. If ka bk D 2, then Œa; bs WD Bd Œ aCb 2 ; 1, where Bd Œp; r denotes the (closed) d -dimensional ball centered at p with radius r in Ed . If ka bk > 2, then we define Œa; bs to be Ed . Next, a set C Ed is called spindle convex if, for any pair of points a; b 2 C, we have that Œa; bs C. Finally, let XTbe a set in Ed . Then the spindle convex hull of X is the set defined by convs X WD fC Ed jX C and C is spindle convex in Ed g. Now, we are ready to state Lemma 6.10.1, which is the core part of this section and whose proof is based on Lemma 6.9.1. Lemma 6.10.1. Let X Ed ; d 3 be a compact set of Euclidean diameter diam.X / 1. Then the boundary of the spindle convex hull of X can be generated as follows: [

bd .convs .X // D

fb C y j y 2 NBŒX .b/g:

b2bd.BŒX /

Proof. Let b 2 bd .BŒX /. Then (ii) of Lemma 6.9.1 implies that b C NBŒX .b/ D Sconv.X \ S d 1 .b; 1/; S d 1 .b; 1// D convs .X / \ S d 1 .b; 1/: This together with the fact that [

NBŒX .b/ D Sd 1

b2bd.BŒX /

finishes the proof of Lemma 6.10.1.

120

6 Proofs on Ball-Polyhedra and Spindle Convex Bodies

6.10.2 On the Euclidean Diameter of Spindle Convex Hulls Lemma 6.10.2. diam .convs .X // 1: Proof. By assumption diam.X / 1. Recall that Meissner [146] has called a compact set M Ed complete if diam.M [ fpg/ > diam.M / for any p 2 Ed n M . He has proved in [146] that any set of diameter 1 is contained in a complete set of diameter 1. Moreover, he has shown in [146] that a compact set of diameter 1 in Ed is complete if and only if it is of constant width 1. These facts together with the easy observation that any convex body of constant width 1 in Ed is in fact a spindle convex set, imply that X is contained in a convex body of convex width 1 and any such convex body must necessarily contain convs .X /. Thus, indeed diam .convs .X // 1. For an arbitrary nonempty subset A of Sd 1 let 0 UBŒX .A/ D @

[

1 NBŒX .b/A Sd 1 :

NBŒX .b/\A¤;

Lemma 6.10.3. Let ; ¤ A Sd 1 be given. Then diam UBŒX .A/ 1 C diam.A/: Proof. Let y1 2 NBŒX .b1 / and y2 2 NBŒX .b2 / be two arbitrary points of UBŒX .A/ with b1 ; b2 2 bd .BŒX /. We need to show that ky1 y2 k 1 C diam.A/. By Lemma 6.10.1 and by Lemma 6.10.2 we get that k.y1 y2 / C .b1 b2 /k D k.b1 C y1 / .b2 C y2 /k 1: Thus, the triangle inequality yields that k.y1 y2 /k 1 C k.b2 b1 /k: This means that in order to finish the proof of Lemma 6.10.3 it is sufficient to show that k.b2 b1 /k diam.A/. This can be obtained easily from the assumption that NBŒX .b1 / \ A ¤ ;; NBŒX .b2 / \ A ¤ ; and from the fact that the sets b1 CNBŒX .b1 / bd .convs .X // and b2 CNBŒX .b2 / bd .convs .X // are separated by the hyperplane H of Ed that bisects the line segment connecting b1 to b2 and is perpendicular to it with b1 C NBŒX .b1 / (resp., b2 C NBŒX .b2 /) lying on the same side of H as b2 (resp., b1 ).

6.10 Proof of Theorem 5.7.3

121

6.10.3 An Upper Bound Based on a Probabilistic Approach Let d 1 denote the standard probability measure on Sd 1 and define Vd 1 .t/ WD inffd 1 .AC / j A Sd 1 ; diam.A/ tg; where just as before AC D fx 2 Sd 1 j hx; yi > 0 for all y 2 Ag. Moreover, let nd 1 ."/ denote the minimum number of closed spherical caps of Sd 1 having Euclidean diameter " such that they cover Sd 1 , where 0 < " 2. Lemma 6.10.4. I.BŒX / 1 C holds for all 0 < "

ln .nd 1 ."// ln .1 Vd 1 .1 C "//

p 2 1 and d 3.

Proof. Let ; ¤ A Sd 1 be given with Euclidean diameter diam.A/ 1 C " p 2. Then the spherical Jung theorem [91] implies that A q is contained in a closed spherical cap of Sd 1 having angular radius 0 < arcsin

contains a spherical cap of Sd 1 having angular radius C

2

d 1 d

arcsin

< q

. 2

d 1 d

Thus, AC

> 0 and of

d 1

course, A is contained in an open hemisphere of S . Hence, 0 < Vd 1 .1 C "/ < 1 2 and so, the expression on the right in Lemma 6.10.4 is well defined. Let m be a positive integer satisfying m>

ln .nd 1 ."// : ln .1 Vd 1 .1 C "//

It is sufficient to show that m directions can illuminate BŒX . Let n D nd 1 ."/ and let A1 ; A2 ; : : : ; An be closed spherical caps of Sd 1 having Euclidean diameter " and covering Sd 1 . By Lemma 6.10.3 we have diam UBŒX .Ai / 1 C " for all 1 i n and therefore d 1 UBŒX .Ai /C Vd 1 .1 C "/ for all 1 i n. Let the directions u1 ; u2 ; : : : ; um be chosen at random, uniformly and independently distributed on Sd 1 . Thus, the probability that uj C lies in UBŒX .Ai / is equal to d 1 UBŒX .Ai /C Vd 1 .1 C "/. Therefore the probability that UBŒX .Ai /C contains none of the points u1 ; u2 ; : : : ; um is at most .1 Vd 1 .1 C "//m . Hence, the probability p that at least one UBŒX .Ai /C will contain none of the points u1 ; u2 ; : : : ; um satisfies p

n X i D1

ln.n/

.1 Vd 1 .1 C "//m < n .1 Vd 1 .1 C "// ln.1Vd 1 .1C"// D 1:

122

6 Proofs on Ball-Polyhedra and Spindle Convex Bodies

This shows that one can choose m directions say, fv1 ; v2 ; : : : ; vm g Sd 1 such that each set UBŒX .Ai /C ; 1 i n contains at least one of them. We claim that the directions v1 ; v2 ; : : : ; vm illuminate BŒX . Indeed, let b 2 bd .BŒX /. We show that at least one of the directions v1 ; v2 ; : : : ; vm illuminates the boundary point b. As the spherical caps A1 ; A2 ; : : : ; An form a covering of Sd 1 therefore there exists an Ai with Ai \ NBŒX .b/ ¤ ;. Thus, by definition NBŒX .b/ UBŒX .Ai / and therefore NBŒX .b/C UBŒX .Ai /C : UBŒX .Ai /C contains at least one of the directions v1 ; v2 ; : : : ; vm , say vk . Hence, vk 2 UBŒX .Ai /C NBŒX .b/C and so, vk illuminates the boundary point b of BŒX , finishing the proof of Lemma 6.10.4.

6.10.4 Schramm’s Lower Bound We need the following notation for the next statement. For u 2 Sd 1 let Ru W Ed ! Ed denote the reflection about the line passing through the points u and u. Clearly, Ru .x/ D 2hx; uiu x for all x 2 Ed . Lemma 6.10.5. Let A Sd 1 be a set of Euclidean diameter 0 < diam.A/ t contained in the closed spherical cap C Œu; arccos a Sd 1 centered at up2 Sd 1 having angular radius 0 < arccos a < 2 with 0 < a < 1 and 0 < t 2 1 a2 . Then 2a ; AC [ Ru .AC / C u; arctan t Sd 1 denotes the open spherical cap centered at u where C u; arctan 2a t having angular radius 0 < arctan. 2a t / < 2. Proof. Suppose that x 2 Sd 1 n AC [ Ru .AC / and let denote the angular distance between x and u. Clearly 0 < and x D .cos /u C .sin /v with v 2 Sd 1 being perpendicular to u. As x … AC (resp., x … Ru .AC / i.e. Ru .x/ … AC ) therefore there exists a point y 2 A (resp., z 2 A) such that 0 hy; ui cos C hy; vi sin .resp:; 0 hz; ui cos hz; vi sin /:

6.10 Proof of Theorem 5.7.3

123

By adding together the last two inequalities and using the inequalities ky zk t and sin 0 we get that 0 hy C z; ui cos C hy z; vi sin hy C z; ui cos t sin : As A C Œu; arccos a Sd 1 therefore if cos > 0, then the last inequality implies that tan Thus, arctan Lemma 6.10.5.

2a t

hy; ui C hz; ui 2a hy C z; ui D : t t t

follows for all 0 < , finishing the proof of

Lemma 6.10.6. 1

Vd 1 .t/ p 8d q for all 0 < t < d2d and d 3. 1

! d 1 2 2 d1 t 2 2 3 C 2 4 2 d2 t 2

Proof. Let ; ¤ A Sd 1 be given with (Euclidean) diameter diam.A/ t. The spherical Jung theorem [91] implies that A is contained in the closed spherical cap q d 1 Sd 1 centered at the properly chosen u 2 Sd 1 having C u; arcsin 2d t q q d 1 2d t < , where by assumption 0 < t < angular radius 0 < arcsin 2d 2 d 1 . Thus, Lemma 6.10.5 implies that 2a AC [ Ru .AC / C u; arctan t with a D

q

1

d 1 2 2d t .

d 1 .AC / D

Hence,

1 1 d 1 .AC / C d 1 .Ru .AC // d 1 AC [ Ru .AC / 2 2

1 1 Svold 1 C u; arctan 2at 2a d 1 C u; arctan D 2 t 2 Svold 1 .Sd 1 / Svold 1 C u; arctan 2a Svold 1 C u; arctan 2at t D D : 2d!d 2d!d

124

6 Proofs on Ball-Polyhedra and Spindle Convex Bodies

As sin arctan. 2a / D 1C t

t2 4a2

12

therefore

2a Svold 1 C u; arctan t 0

2

31 1 2 2 t 2a 5A u; 1 C 2 > vold 1 @Bd 1 4cos arctan t 4a d 1 d 1 2 2 t2 !d 1 t2 C 1C 2 !d 1 and so; d 1 .A / : D 1C 2 4a 2d!d 4a Hence, using the well-known estimate (see also [167]) 1 d 1 .A / 2d

r

C

Finally, substituting a D

q

1

d 1 2 2d t

d 1 .AC / p

1

D p 8d

d 2

!d 1 !d

q

d 2

we get that

d 1 2 t2 1C 2 : 4a

we are led to the following inequality

1 8d

1C

t2 4

! d 1 2

2.d 1/t 2 d

! d 1 2 2 d1 t 2 2 3 C : 2 4 2 d2 t 2

This finishes the proof of Lemma 6.10.6.

6.10.5 On Sets of Given Diameter to Cover Spherical Space Lemma 6.10.7. 4 d nd 1 ."/ < 1 C " for all 0 < " 2 and d 3.

6.10 Proof of Theorem 5.7.3

125

Proof. Let fp1 ; p2 ; : : : pn g Sd 1 be the largest family of points Sd 1 with the Son n " property that kpi pj k 2 for all 1 i < j n. Then clearly i D1 Bd pi ; 2" Sd 1 and therefore n nd 1 ."/. As the balls Bd Œpi ; 4" ; 1 i n form a packing in Bd Œo; 1 C 4" therefore n

" d 4

" d !d < 1 C !d ; 4

implying that

1C " nd 1 ."/ n < d4 "

d

4

4 d D 1C : "

This completes the proof of Lemma 6.10.7.

Actually, using [94], one can replace the inequality of Lemma 6.10.7 by the d stronger inequality nd 1 ."/ . 21 C o.1//d ln d 2" . As this improves the estimate of Theorem 5.7.3 only in a rather insignificant way, we do not introduce it here.

6.10.6 The Final Upper Bound for the Illumination Number Now, we are ready for the proof of Theorem 5.7.3. As x < ln.1 x/ holds for all 0 < x < 1, therefore by Lemma 6.10.4 we get that I.BŒX / 1 C

ln .nd 1 ."// ln .nd 1 ."// <1C ln .1 Vd 1 .1 C "// Vd 1 .1 C "/

q p holds for all 0 < " 2 1 and d 3. Now, let "0 D 2d2d1 1. As 0 < "0 < p 2 1 holds for all d 3, therefore Lemmas 6.10.6 and 6.10.7 together with the easy inequality "0 > 16d41 yield that p 3 I.BŒX / < 1 C 8d 2 p 3 < 1 C 8d 2 r D1C4

d 1 2

p d d 3

d 1 2

ln .nd 1 ."0 //

! d 1 p 4 d 3 2 < 1 C 8d ln 1C ln .16d /d "0 2

d2 d2 12 3 3 3 2 .ln 16 C ln d / < 4 d .3 C ln d / ; 2 3 2

finishing the proof of Theorem 5.7.3.

126

6 Proofs on Ball-Polyhedra and Spindle Convex Bodies

6.11 Proof of Theorem 5.7.5 Theorem 5.7.5. Let 0 < w < be given. Then the volume of the convex body C of constant width w in S3 is minimal among all convex bodies of constant width w of S3 if and only if the polar body C of constant width w has minimal volume among all convex bodies of constant width w of S3 . The following is a short proof, but one cannot call it trivial. Let C 2 KS3 be an arbitrary convex body of constant width 0 < w < with sufficiently smooth boundary in S3 . On the one hand, according to a classical result of Blaschke [61] we have that M1 .C/ C 2V .C/ D 2w; where M1 .C/ is the integral of the mean curvature (evaluated over the boundary of C) and V .C/ denotes the (spherical) volume of C. (See also formula (5.7) in [164].) On the other hand, Allendoerfer [6] has proved that (for not only the above C, but actually, for any C 2 KS3 with sufficiently smooth boundary) we have also M1 .C/ C V .C/ C V .C / D 2 : (See also formula (17.31) in [165].) Clearly, the above two equations imply that 2 C V .C/ D 2w C V .C / holds for any convex body C of constant width 0 < w < in S3 , from which Theorem 5.7.5 follows in a straightforward way.

6.12 Proof of Theorem 5.8.1 Theorem 5.8.1. Let be the set containing all faces of a standard ball-polyhedron P Ed and the empty set and P itself. Then is a finite bounded lattice with respect to ordering by inclusion. The atoms of are the vertices of P and is atomic; that is, for every element F 2 with F ¤ ; there is a vertex v of P such that v 2 F . Moreover, P has k-dimensional faces for every 0 k d 1 and P is the spindle convex hull of its .d 2/-dimensional faces. Furthermore, no standard ball-polyhedron in Ed is the spindle convex hull of its .d 3/-dimensional faces. d C1 Finally, D Pd 1 if fii .P/ denotes the number of i -dimensional faces of P, then 1C.1/ i D0 .1/ fi .P/.

6.12 Proof of Theorem 5.8.1

127

6.12.1 The CW-Decomposition of a Standard Ball-Polyhedron Let K be a convex body in Ed and b 2 bdK. Then recall that the Gauss image of b with respect to K is the set of outward unit normal vectors of hyperplanes that support K at b. Clearly, it is a spherically convex subset of S d 1 .o; 1/ and its dimension is defined in the natural way. Theorem 6.12.1. Let P be a standard ball-polyhedron. Then the faces of P form the closed cells of a finite CW-decomposition of the boundary of P. Proof. Let fS d 1 .p1 ; 1/; : : : ; S d 1 .pk ; 1/g be the reduced family of generating spheres of P. The relative interior (resp., the relative boundary) of an m-dimensional face F of P is defined as the set of those points of F that are mapped to Bm .o; 1/ (resp., S m1 .o; 1/) under any homeomorphism between F and Bm Œo; 1. For every b 2 bdP define the following sphere S.b/ WD

\

fS d 1 .pi ; 1/ W pi 2 S d 1 .b; 1/; i 2 f1; : : : ; kgg:

Clearly, S.b/ is a support sphere of P. Moreover, if S.b/ is an m-dimensional sphere, then the face F WD S.b/ \ P is also m-dimensional as b has an mdimensional neighbourhood in S.b/ that is contained in F . This also shows that b belongs to the relative interior of F . Hence, the union of the relative interiors of the faces covers bdP. We claim that every face F of P can be obtained in this way; that is, for any relative interior point b of F we have F D S.b/ \ P. Clearly, F S.b/ \ P, as the support sphere of P that intersects P in F contains S.b/. It is sufficient to show that F is at most m-dimensional. This is so, because the Gauss image of b with respect to at least .d m 1/-dimensional, since the Gauss image of b with respect to T P is fBd Œpi ; 1 W pi 2 S d 1 .b; 1/; i 2 f1; : : : ; kgg P is .d m 1/-dimensional. The above argument also shows that no point b 2 bdP belongs to the relative interior of more than one face. Moreover, if b 2 bdP is on the relative boundary of the face F then S.b/ is clearly of smaller dimension than F . Hence, b belongs to the relative interior of a face of smaller dimension. This concludes the proof of Theorem 6.12.1.

6.12.2 On the Number of Generating Balls Corollary 6.12.2. The generating balls of any standard ball-polyhedron P in Ed consist of at least d C 1 unit balls. Proof. Because the faces form a CW-decomposition of the boundary of P, there is a vertex v. The Gauss image of v is .d 1/-dimensional. So, v belongs to at least d generating spheres from the family of generating balls. We denote the

128

6 Proofs on Ball-Polyhedra and Spindle Convex Bodies

centers of those spheres by x1 ; x2 ; : : : ; xd . Let H WD afffx1 ; x2 ; : : : ; xd g. Then BŒfx1 ; x2 ; : : : ; xd g, which denotes the intersection of the closed d -dimensional unit balls centered at the points x1 ; x2 ; : : : ; xd , is symmetric about H . Let H be the reflection of Ed about H . Then S WD S d 1 .x1 ; 1/ \ S d 1 .x2 ; 1/ \ \ S d 1 .xd ; 1/ contains the points v and H .v/, hence S is a sphere, not a point. Finally, as P is a standard ball-polyhedron, therefore there is a unit-ball Bd Œxd C1 ; 1 in the family of generating balls of P that does not contain S .

6.12.3 On the Face Lattices of Standard Ball-Polyhedra Corollary 6.12.3. Let be the set containing all faces of a standard ballpolyhedron P Ed and the empty set and P itself. Then is a finite bounded lattice with respect to ordering by inclusion. The atoms of are the vertices of P and is atomic: for every element F 2 with F ¤ ; there is a vertex x of P such that x 2 F . Proof. First, we show that the intersection of two faces F1 and F2 is another face (or the empty set). The intersection of the two supporting spheres that intersect P in F1 and F2 is another supporting sphere of P, say S l .p; r/. Then S l .p; r/ \ P D F1 \ F2 is a face of P. From this the existence of a unique maximum common lower bound (i.e., an infimum) for F1 and F2 follows. Moreover, by the finiteness of , the existence of a unique infimum for any two elements of implies the existence of a unique minimum common upper bound (i.e., a supremum) for any two elements of , say C and D, as follows. The supremum of C and D is the infimum of all the (finitely many) elements of that are above C and D. Vertices of P are clearly atoms of . Using Theorem 6.12.1 and induction on the dimension of the face it is easy to show that every face is the supremum of its vertices. Corollary 6.12.4. A standard ball-polyhedron P in Ed has k-dimensional faces for every 0 k d 1. Proof. We use an inductive argument on k, where we go from k D d 1 down to k D 0. Clearly, P has facets. A k-face F of P is homeomorphic to Bk Œo; 1, hence its relative boundary is homeomorphic to S k1 .o; 1/, if k > 0. Since the .k 1/skeleton of P covers the relative boundary of F , P has .k 1/-faces. Corollary 6.12.5. Let d 3. Any standard ball-polyhedron P is the spindle convex hull of its .d 2/-dimensional faces. Furthermore, no standard ball-polyhedron is the spindle convex hull of its .d 3/-dimensional faces. Proof. For the first statement, it is sufficient to show that the spindle convex hull of the .d 2/-faces contains the facets. Let p be a point on the facet, F D P \ S d 1 .q; 1/. Take any great circle C of S d 1 .q; 1/ passing through p. Since F is

6.12 Proof of Theorem 5.8.1

129

spherically convex on S d 1 .q; 1/, C \ F is a unit circular arc of length less than . Let r; s 2 S d 1 .q; 1/ be the two endpoints of C \ F . Then r and s belong to the relative boundary of F . Hence, by Theorem 6.12.1, r (resp., s) belongs to a .d 2/face. Clearly, p 2 convs fr; sg. The proof of the second statement goes as follows. By Corollary 6.12.4 we can choose a relative interior point p of a .d 2/-dimensional face F of P. Let q1 and q2 be the centers of the generating balls of P such that F WD S d 1 .q1 ; 1/ \ S d 1 .q2 ; 1/ \ P. Clearly, p … convs ..Bd Œq1 ; 1 \ Bd Œq2 ; 1/nfpg/ convs .Pnfpg/. Corollary 6.12.6 (Euler–Poincar´e Formula). If P is an arbitrary standard d -dimensional ball-polyhedron, then 1 C .1/d C1 D

d 1 X

.1/i fi .P/;

i D0

where fi .P/ denotes the number of i -dimensional faces of P. Proof. It follows from Theorem 6.12.1 and the fact that a ball-polyhedron in Ed is a convex body, hence its boundary is homeomorphic to S d 1 .o; 1/.

Chapter 7

Coverings by Cylinders

Abstract In the 1930s, A.Tarski introduced his plank problem at a time when the field discrete geometry was about to born. It is quite remarkable that Tarski’s question and its variants continue to generate interest in the geometric and analytic aspects of coverings by cylinders in the present time as well. This chapter surveys plank theorems, covering convex bodies by cylinders, Kadets–Ohmann-type theorems and investigates partial coverings of balls by planks.

7.1 Plank Theorems: Old and New As usual, a convex body of the Euclidean space Ed is a compact convex set with nonempty interior. Let C Ed be a convex body, and let H Ed be a hyperplane. Then the distance w.C; H / between the two supporting hyperplanes of C parallel to H is called the width of C parallel to H . Moreover, the smallest width of C parallel to hyperplanes of Ed is called the minimal width of C and is denoted by w.C/. Recall that in the 1930s, Tarski posed what came to be known as the plank problem. A plank P in Ed is the (closed) set of points between two distinct parallel hyperplanes. The width w.P/ of P is simply the distance between the two boundary hyperplanes of P. Tarski conjectured that if a convex body of minimal width w is covered by a collection of planks in Ed , then the sum of the widths of these planks is at least w. This conjecture was proved by Bang in his memorable paper [15]. (In fact, the proof presented in that paper is a simplification and generalization of the proof published by Bang somewhat earlier in [14].) Thus, we call the following statement Bang’s plank theorem. Theorem 7.1.1. If the convex body C is covered by the planks P1 ; P2 ; : : : ; Pn in Ed ; d 2 (i.e., C P1 [ P2 [ [ Pn Ed ), then n X

w.Pi / w.C/:

i D1

K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 7, © Springer International Publishing Switzerland 2013

131

132

7 Coverings by Cylinders

In [15], Bang raised the following stronger version of Tarski’s plank problem called the affine plank problem. We phrase it via the following definition. Let C be a convex body and let P be a plank with boundary hyperplanes parallel to the w.P/ hyperplane H in Ed . We define the C-width of the plank P as w.C;H / and label it wC .P/. (This notion was introduced by Bang [15] under the name “relative width”.) Conjecture 7.1.2. If the convex body C is covered by the planks P1 ; P2 ; : : : ; Pn in Ed ; d 2, then n X

wC .Pi / 1:

i D1

The special case of Conjecture 7.1.2, when the convex body to be covered is centrally symmetric, has been proved by Ball in [12]. Thus, the following is Ball’s plank theorem. Theorem 7.1.3. If the centrally symmetric convex body C is covered by the planks P1 ; P2 ; : : : ; Pn in Ed ; d 2, then n X

wC .Pi / 1:

i D1

From the point of view of discrete geometry it seems natural to mention that after proving Theorem 7.1.3 Ball [13] used Bang’s proof of Theorem 7.1.1 to derive a new argument for an improvement of the Davenport–Rogers lower bound on the density of economical sphere lattice packings. It was Alexander [3] who noticed that Conjecture 7.1.2 is equivalent to the following generalization of a problem of Davenport. Conjecture 7.1.4. If a convex body C in Ed ; d 2 is sliced by n 1 hyperplane cuts, then there exists a piece that covers a translate of n1 C. We note that the paper [45] of A. Bezdek and the author proves Conjecture 7.1.4 for successive hyperplane cuts (i.e., for hyperplane cuts when each cut divides one piece). Also, the same paper [45] introduced two additional equivalent versions of Conjecture 7.1.2. As they seem to be of independent interest we recall them following the terminology used in [45]. Let C and K be convex bodies in Ed and let H be a hyperplane of Ed . The / C-width of K parallel to H is denoted by wC .K; H / and is defined as w.K;H . w.C;H / The minimal C-width of K is denoted by wC .K/ and is defined as the minimum of wC .K; H /, where the minimum is taken over all possible hyperplanes H of Ed . Recall that the inradius of K is the radius of the largest ball contained in K. It is quite natural then to introduce the C-inradius of K as the factor of the largest positive homothetic copy of C, a translate of which is contained in K. We need to do one more step to introduce the so-called successive C-inradii of K as follows.

7.1 Plank Theorems: Old and New

133

Fig. 7.1 The C-rounded body of K: K C

Fig. 7.2 The 3rd successive C-inradius of K: rC .K; 3/ D with the characteristic property wC .K C / D 3

Let r be the C-inradius of K. For any 0 < r let the C-rounded body of K be denoted by K C and be defined as the union of all translates of C that are covered by K. (See Fig. 7.1.) Now, take a fixed integer m 1. On the one hand, if > 0 is sufficiently small, then wC .K C / > m . On the other hand, wC .KrC / D r mr. As wC .K C / is a decreasing continuous function of > 0 and m is a strictly increasing continuous function of , there exists a uniquely determined > 0 such that wC .K C / D m : This uniquely determined is called the mth successive C-inradius of K and is denoted by rC .K; m/. (See Fig. 7.2.) Notice that rC .K; 1/ D r. For the sake of completeness we give a somewhat different but still equivalent description of rC .K; m/. If C is a convex body in Ed , then t C C; t C 2 v C C; : : : ; t C m v C C is called a linear packing of m translates of C positioned parallel to the line f v j 2 Rg with direction vector v ¤ o if the m translates of C are pairwise non-overlapping, i.e., if .t C i v C intC/ \ .t C j v C intC/ D ; holds for all 1 i ¤ j m (with 1 D 0). Furthermore, the line l Ed passing through the origin o of Ed is called a separating direction for the linear packing t C C; t C 2 v C C; : : : ; t C m v C C

134

7 Coverings by Cylinders

if Prl .t C C/; Prl .t C 2 v C C/; : : : ; Prl .t C m v C C/ are pairwise non-overlapping intervals on l, where Prl W Ed ! l denotes the orthogonal projection of Ed onto l. It is easy to see that every linear packing t C C; t C 2 v C C; : : : ; t C m v C C possesses at least one separating direction in Ed . Finally, let K be a convex body in Ed and let m 1 be a positive integer. Then let > 0 be the largest positive real with the following property: for every line l passing through the origin o in Ed there exists a linear packing of m translates of C lying in K and having l as a separating direction. It is straightforward to show that

D rC .K; m/: Now, the two equivalent versions of Conjectures 7.1.2 and 7.1.4 introduced in [45] can be phrased as follows. d Conjecture 7.1.5. Pn If a convex body K in E ; d 2 is covered by dthe planks P1 ; P2 ; : : : ; Pn , then i D1 wC .Pi / wC .K/ for any convex body C in E .

Conjecture 7.1.6. Let K and C be convex bodies in Ed ; d 2. If K is sliced by n 1 hyperplanes, then the minimum of the greatest C-inradius of the pieces is equal to the nth successive C-inradius of K, that is, it is rC .K; n/. Recall that Theorem 7.1.3 gives a proof of (Conjecture 7.1.5 as well as) Conjecture 7.1.6 for centrally symmetric convex bodies K in Ed ; d 2 (with C being an arbitrary convex body in Ed ; d 2). Another approach that leads to a partial solution of Conjecture 7.1.6 was published in [45]. Namely, in that paper A. Bezdek and the author proved the following theorem that (under the condition that C is a ball) answers a question raised by Conway [44] as well as proves Conjecture 7.1.6 for successive hyperplane cuts. Theorem 7.1.7. Let K and C be convex bodies in Ed , d 2. If K is sliced into n 1 pieces by n 1 successive hyperplane cuts (i.e., when each cut divides one piece), then the minimum of the greatest C-inradius of the pieces is the nth successive C-inradius of K (i.e., rC .K; n/). An optimal partition is achieved by n 1 parallel hyperplane cuts equally spaced along the minimal C-width of the rC .K; n/C-rounded body of K. Next we state the following stronger version of Theorem 7.1.7. Its proof is a natural extension of the proof of Theorem 7.1.7 published in [45]. Theorem 7.1.8. Let K and C be convex bodies in Ed , d 2 and let m be a positive integer. If K is sliced into n 1 pieces by n 1 successive hyperplane cuts (i.e., when each cut divides one piece), then the minimum of the greatest mth successive

7.1 Plank Theorems: Old and New

135

C-inradius of the pieces is the .mn/th successive C-inradius of K (i.e., rC .K; mn/). An optimal partition is achieved by n 1 parallel hyperplane cuts equally spaced along the minimal C-width of the rC .K; mn/C-rounded body of K. Theorem 7.1.8 leads us to the following general but still equivalent version of Conjecture 7.1.6. Conjecture 7.1.9. Let K and C be convex bodies in Ed ; d 2 and let the positive integer m be given. If K is sliced by n 1 hyperplanes, then the minimum of the greatest mth successive C-inradius of the pieces is the .mn/th successive C-inradius of K, that is, it is rC .K; mn/. It is immediate to observe that also Conjecture 7.1.5 has a general version which is in fact, an equivalent one. In order to see that the following conjecture is equivalent to Conjecture 7.1.5 it is sufficient to note that the sequence mrC .K; m/; m D 1; 2; : : : is an increasing one with lim mrC .K; m/ D wC .K/:

m!C1

Conjecture 7.1.10. Let K and C be convex bodies in Ed ; d 2 and let m be P1 ; P2 ; : : : ; Pn in Ed , then Pna positive integer. If K is covered by the Pplanks n i D1 rC .Pi ; m/ rC .K; m/ or equivalently, i D1 wC .Pi / mrC .K; m/. Akopyan and Karasev [2] just very recently have proved the following partial result on Conjecture 7.1.10. Their result is based on a generalization of successive hyperplane cuts. The more exact details are as follows. Under the convex partition V1 [ V2 [ [ Vn of Ed we understand the family V1 ; V2 ; : : : ; Vn of closed convex sets having pairwise disjoint non-empty interiors in Ed with V1 [V2 [ [Vn D Ed . Then we say that the convex partition V1 [ V2 [ [ Vn of Ed is an inductive partition of Ed if for any 1 i n, there exists an inductive partition W1 [ [ Wi 1 [Wi C1 [ [Wn of Ed such that Vj Wj for all j ¤ i . A partition into one part V1 D Ed is assumed to be inductive. We note that if Ed is sliced into n pieces by n 1 successive hyperplane cuts (i.e., when each cut divides one piece), then the pieces generate an inductive partition of Ed . Also, the Voronoi cells of finitely many points of Ed generate an inductive partition of Ed . Now, the main theorem of [2] can be phrased as follows. Theorem 7.1.11. Let K and C be convex bodies in Ed ; d 2 and let V1 [ V2 [ [ Vn be an inductive partition of Ed such that int.Vi \ K/ ¤ ; for all 1 i n. Then n X

rC .Vi \ K; 1/ rC .K; 1/:

i D1

In fact, the method of Akopyan and Karasev [2] can be extended to prove the following stronger version of Theorem 7.1.11.

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Theorem 7.1.12. Let K and C be convex bodies in Ed ; d 2 and let m be a positive integer. If V1 [ V2 [ [ Vn is an inductive partition of Ed such that int.Vi \ K/ ¤ ; for all 1 i n, then n X

rC .Vi \ K; m/ rC .K; m/:

i D1

7.2 Covering Convex Bodies by Cylinders In his paper [15], Bang by describing a concrete example and writing that it may be extremal proposes to investigate a quite challanging question that can be phrased as follows. Problem 7.2.1. Prove or disprove that the sum of the base areas of finitely many cylinders covering a 3-dimensional convex body is at least half of the minimum area 2-dimensional projection of the body. If true, then the estimate of Problem 7.2.1 is a sharp one due to a covering of a regular tetrahedron by two cylinders described in [15]. A very recent paper of Litvak and the author [51] investigates Problem 7.2.1 as well as its higher dimensional analogue. Their main result can be summarized as follows. Given 0 < k < d define a k-codimensional cylinder C in Ed as a set which can be presented in the form C D H C B, where H is a k-dimensional linear subspace of Ed and B is a measurable set (called the base) in the orthogonal complement H ? of H . For a given convex body K and a k-codimensional cylinder C D H C B we define the cross-sectional volume crvK .C/ of C with respect to K as follows crvK .C/ WD

vol d k .PH ? C/ vol d k .B/ vol d k .C \ H ? / D D ; vol d k .PH ? K/ vol d k .PH ? K/ vol d k .PH ? K/

where PH ? W Ed ! H ? denotes the orthogonal projection of Ed onto H ? . Notice that for every invertible affine map T W Ed ! Ed one has crvK .C/ D crvT K .T C/. The following theorem is proved in [51]. Theorem 7.2.2. Let K be a convex body in S Ed . Let C1 ; : : : ; CN be k-codimensional d cylinders in E ; 0 < k < d such that K N i D1 Ci : Then N X i D1

1 crvK .Ci / d : k

Moreover, if K S is an ellipsoid and C1 ; : : : ; CN are 1-codimensional cylinders in Ed such that K N i D1 Ci , then

7.3 Kadets–Ohmann-Type Theorems

137

N X

crvK .Ci / 1:

i D1

The case k D d 1 of Theorem 7.2.2 corresponds to Conjecture 7.1.2 i.e. to the affine plank problem. Theorem 7.2.2 for k D d 1 implies the lower bound 1=d that can be somewhat further improved (for more details see [51]). As an immediate corollary of Theorem 7.2.2 we get the following estimate for Problem 7.2.1. Corollary 7.2.3. The sum of the base areas of finitely many (1-codimensional) cylinders covering a 3-dimensional convex body is always at least one third of the minimum area 2-dimensional projection of the body. Note that the inequality of Theorem 7.2.2 on covering ellipsoids by 1codimensional cylinders is best possible. By looking at this result from the point of view of k-codimensional cylinders we are led to ask the following quite natural question. Unfortunately, despite its elementary character it is still open. Problem 7.2.4. Let 0 < c.d; k/ 1 denote the largest real number with the k-codimensional cylinders property that if K is an ellipsoid and CS 1 ; : : : ; CN are P N in Ed ; 1 k d 1 such that K N C , then i D1 i i D1 crvK .Ci / c.d; k/: Determine c.d; k/ for given d and k. Clearly, Theorems 7.1.1 and 7.2.2 imply that c.d; d 1/ D 1 and c.d; 1/ D 1 moreover, c.d; k/ 1d . .k /

7.3 Kadets–Ohmann-Type Theorems Recall that Ball [12] generalized the plank theorem of Bang [14, 15] for coverings of balls by planks in Banach spaces (where planks are defined with the help of linear functionals instead of inner product). This theorem was further strengthened by Kadets [134] for real Hilbert spaces as follows. Let C be a closed convex subset with non-empty interior in the real Hilbert space H (finite or infinite dimensional). We call C a convex body of H. Then let r.C/ denote the supremum of the radii of the balls contained in C. (One may call r.C/ the inradius of C.) Planks and their widths in H are defined with the help of the inner product of H in the usual way. Thus, if C is a convex body in H and P is a plank of H, then the width w.P/ of P is always at least as large as 2r.C \ P/. Now, the main result of [134] is the following. Theorem 7.3.1. Let the ball B of the real Hilbert space H be covered by the convex bodies C1 ; C2 ; : : : ; Cn in H. Then n X i D1

r.Ci \ B/ r.B/:

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7 Coverings by Cylinders

We note that the 2-dimensional Euclidean case of Theorem 7.3.1 had been stated and proved by Ohmann [154] several years before the publication of [134] (see also the recent paper [36] of A. Bezdek for reproving that 2-dimensional case independently). Kadets [134] proposes to investigate the analogue of Theorem 7.3.1 in Banach spaces. Thus, an affirmative answer to the following problem would improve the plank theorem of Ball. Problem 7.3.2. Let the ball B be covered by the convex bodies C1 ; C2 ; : : : ; Cn in an arbitrary Banach space. Prove or disprove that n X

r.Ci \ B/ r.B/:

i D1

Next, we discuss an extension of Theorem 7.3.1 to coverings of large balls in spherical spaces proved just recently by Schneider and the author in [55]. Recall that Sd stands for the d -dimensional unit sphere in the .d C 1/-dimensional Euclidean space Ed C1 ; d 2. A spherically convex body is a closed, spherically convex subset K of Sd with interior points and lying in some closed hemisphere, thus, the intersection of Sd with a .d C 1/-dimensional closed convex cone of Ed C1 different from Ed C1 . The inradius r.K/ of K is the spherical radius of the largest spherical ball contained in K. Also, recall that a lune in Sd is the d -dimensional intersection of Sd with two closed halfspaces of Ed C1 with the origin o in their boundaries. The intersection of the boundaries (or any .d 1/-dimensional subspace in that intersection, if the two subspaces are identical) is called the ridge of the lune. Evidently, the inradius of a lune is half the interior angle between the two defining hyperplanes. Theorem 7.3.3. If the spherically convex bodies K1 ; : : : ; Kn cover the spherical ball B of radius r.B/ 2 in Sd ; d 2, then n X

r.Ki / r.B/:

i D1

P For r.B/ D 2 the stronger inequality niD1 r.Ki \ B/ r.B/ holds. Moreover, equality for r.B/ D or r.B/ D 2 holds if and only if K1 ; : : : ; Kn are lunes with common ridge which have pairwise no common interior points. Theorem 7.3.3 is a consequence of the following result proved by Schneider and the author in [55]. Recall that Svold .: : : / denotes the spherical Lebesgue measure on Sd , and recall that .d C 1/!d C1 D Svold .Sd /. Theorem 7.3.4. If K is a spherically convex body in Sd ; d 2, then Svold .K/

.d C 1/!d C1 r.K/:

Equality holds if and only if K is a lune.

7.4 On Partial Coverings of Balls by Planks

139

Indeed, Theorem 7.3.4 implies Theorem 7.3.3 as follows. If B D Sd ; that is, the spherically convex bodies K1 ; : : : ; Kn cover Sd , then .d C 1/!d C1

n X

Svold .Ki /

i D1

n .d C 1/!d C1 X r.Ki /; i D1

and the stated inequality follows. In general, when B is different from Sd , let B0 Sd be the spherical ball of radius r.B/ centered at the point antipodal to the center of B. As the spherically convex bodies B0 ; K1 ; : : : ; Kn cover Sd , the inequality just proved shows that r.B/ C

n X

r.Ki / ;

i D1

and the stated inequality follows. If r.B/ D 2 , then K1 \ B; : : : ; Kn \ B are spherically convex bodies and as B0 ; K1 \ B; : : : ; Kn \ B cover Sd , the stronger inequality follows. The assertion about the equality sign for the case when r.B/ D or r.B/ D 2 follows easily. We close this section with the following question that bridges Theorems 7.3.3 to 7.3.1: Problem 7.3.5. Let the spherically convex bodies K1 ; : : : ; Kn cover the spherical ball B of radius r.B/ < 2 in Sd ; d 2. Then prove or disprove that n X

r.Ki / r.B/:

i D1

7.4 On Partial Coverings of Balls by Planks The following variant of Tarski’s plank problem was introduced very recently by the author in [38]: let C be a convex body of minimal width w > 0 in Ed . Moreover, let w1 > 0; w2 > 0; : : : ; wn > 0 be given with w1 C w2 C C wn < w. Then find the arrangement of n planks say, of P1 ; P2 ; : : : ; Pn , of width w1 ; w2 ; : : : ; wn in Ed such that their union covers the largest volume subset of C, that is, for which vold ..P1 [ P2 [ [ Pn / \ C/ is as large as possible. As the following special case is the most striking form of the above problem, we are putting it forward as the main question of this section. Problem 7.4.1. Let Bd denote the unit ball centered at the origin o in Ed . Moreover, let w1 ; w2 ; : : : ; wn be positive real numbers satisfying the inequality w1 C w2 C C wn < 2. Then prove or disprove that the union of the planks P1 ; P2 ; : : : ; Pn of width w1 ; w2 ; : : : ; wn in Ed covers the largest volume subset of Bd if and only if P1 [ P2 [ [ Pn is a plank of width w1 C w2 C C wn with o as a center of symmetry.

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7 Coverings by Cylinders

Clearly, there is an affirmative answer to Problem 7.4.1 for n D 1. Also, we note that it would not come as a surprise to us if it turned out that the answer to Problem 7.4.1 is positive in proper low dimensions and negative in (sufficiently) high dimensions. The following partial results have been obtained in [38]. Theorem 7.4.2. Let w1 ; w2 ; : : : ; wn be positive real numbers satisfying the inequality w1 C w2 C C wn < 2. Then the union of the planks P1 ; P2 ; : : : ; Pn of width w1 ; w2 ; : : : ; wn in E3 covers the largest volume subset of B3 if and only if P1 [ P2 [ [ Pn is a plank of width w1 C w2 C C wn with o as a center of symmetry. Corollary 7.4.3. If P1 ; P2 , and P3 are planks in Ed , d 3 of widths w1 ; w2 , and w3 satisfying 0 < w1 C w2 C w3 < 2, then P1 [ P2 [ P3 covers the largest volume subset of Bd if and only if P1 [ P2 [ P3 is a plank of width w1 C w2 C w3 having o as a center of symmetry. The following estimate of [38] can be derived from Bang’s paper [15]. In order to state it properly we introduce two definitions. m Let C be a convex body in Ed and let m be a positive integer. Then let TC;d denote the family of all sets in Ed that can be obtained as the intersection of at most m translates of C in Ed . Let C be a convex body of minimal width w > 0 in Ed and let 0 < x w be given. Then for any non-negative integer n let n

2 and w.Q/ x g: vd .C; x; n/ WD minfvold .Q/ j Q 2 TC;d

Now, we are ready to state the theorem which was not published by Bang in [15], still it follows from his proof of Tarski’s plank conjecture. Theorem 7.4.4. Let C be a convex body of minimal width w > 0 in Ed . Moreover, let P1 ; P2 ; : : : ; Pn be planks of width w1 ; w2 ; : : : ; wn in Ed with w0 D w1 C w2 C C wn < w. Then vold .C n .P1 [ P2 [ [ Pn // vd .C; w w0 ; n/; i.e.; vold ..P1 [ P2 [ [ Pn / \ C/ vold .C/ vd .C; w w0 ; n/: Clearly, the first inequality above implies (via an indirect argument) that if the planks P1 ; P2 ; : : : ; Pn of width w1 ; w2 ; : : : ; wn cover the convex body C in Ed , then w1 C w2 C C wn w. Also, as an additional observation from [38] we mention the following statement, that can be derived from Theorem 7.4.4 in a straightforward way and, on the other hand, represents the only case when the estimate in Theorem 7.4.4 is sharp.

7.4 On Partial Coverings of Balls by Planks

141

Corollary 7.4.5. Let T be an arbitrary triangle of minimal width (i.e., of minimal height) w > 0 in E2 . Moreover, let w1 ; w2 ; : : : ; wn be positive real numbers satisfying the inequality w1 C w2 C C wn < w. Then the union of the planks P1 ; P2 ; : : : ; Pn of width w1 ; w2 ; : : : ; wn in E2 covers the largest area subset of T if P1 [ P2 [ [ Pn is a plank of width w1 C w2 C C wn sitting on the side of T with height w. It was observed by the author in [38] that there is an implicit connection between Problem 7.4.1 and the well-known Blaschke–Lebesgue problem, which is generated by Theorem 7.4.4. The details are as follows. First, recall that the Blaschke–Lebesgue problem is about finding the minimum volume convex body of constant width w > 0 in Ed . In particular, the Blaschke– Lebesgue theorem states that among all convex domains of constant p width w, the Reuleaux triangle of width w has the smallest area, namely 12 . 3/w2 . Blaschke [60] and Lebesgue [143] were the first to show this and the succeeding decades have seen other works published on different proofs of that theorem. For a most recent new proof, and for a survey on the state of the art of different proofs of the Blaschke–Lebesgue theorem, see the elegant paper of Harrell [125]. Here we note that the Blaschke–Lebesgue problem is unsolved in three and more dimensions. Even finding the 3-dimensional set of least volume presents formidable difficulties. On the one hand, Chakerian [74] pproved that any convex body of constant width 1 in E3 has volume at least .3 367/ D 0:365 : : :. On the other hand, it has been conjectured by Bonnesen and Fenchel [65] that Meissner’s p 3-dimensional generalizations of the Reuleaux triangle of volume . 23 14 3 arccos. 13 // D 0:420 : : : are the only extramal sets in E3 . w;d For our purposes it is useful to introduce the notation Kw;d BL (resp., KBL ) for a convex body of constant width w in Ed having minimum volume (resp., surface w;d volume). One may call Kw;d BL (resp., KBL ) a Blaschke–Lebesgue-type convex body with respect to volume (resp., surface volume). Note that for d D 2; 3 one may w;d choose Kw;d BL D KBL , however, this is likely not to happen for d 4. (For more details on this see [74].) As an important note we mention that Schramm [168] has proved the inequality r vold .Kw;d BL /

2 1 3C d C1

d d w vold .Bd /; 2

which gives the best lower bound for all d > 4. By observing that the orthogonal projection of a convex body of constant width w in Ed onto any hyperplane of Ed is a .d 1/-dimensional convex body of constant width w one obtains from the previous inequality of Schramm the following one, w;d svold 1 .bd.KBL //

r d 1 d 1 2 w d 3C 1 vold .Bd /: d 2

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7 Coverings by Cylinders

Second, let us recall that if X is a finite (point) set lying in the interior of a unit ball in Ed , then the intersection of the (closed) unit balls of Ed centered at the points of X is called a ball-polyhedron and it is denoted by BŒX . (For an extensive list of properties of ball-polyhedra see the recent paper [59].) Now, we are ready to state our theorem. Theorem 7.4.6. Let BŒX Ed be a ball-polyhedron of minimal width x with 1 x < 2. Then 2x;d

2x;d vold .BŒX / vold .KBL / C svold 1 .bd.KBL

//.x 1/ C vold .Bd /.x 1/d :

Thus, Theorems 7.4.4 and 7.4.6 imply the following. Corollary 7.4.7. Let Bd denote the unit ball centered at the origin o in Ed , d 2. Moreover, let P1 ; P2 ; : : : ; Pn be planks of width w1 ; w2 ; : : : ; wn in Ed with w0 D w1 C w2 C C wn 1. Then vold ..P1 [ P2 [ [ Pn / \ Bd / vold .Bd / vd .Bd ; 2 w0 ; n/ w0 ;d

w0 ;d .1 .1 w0 /d /vold .Bd / vold .KBL / svold 1 .bd.KBL //.1 w0 /:

Chapter 8

Proofs on Coverings by Cylinders

Abstract First, we give a proof of the long-standing affine plank conjecture of Bang for successive hyperplane cuts and then for inductive partitions. Second, we prove a lower estimate for the sum of the cross-sectional volumes of cylinders covering a convex body in Euclidean d -space. Then we prove a Kadets–Ohmann-type theorem in spherical d -space for coverings of balls by convex bodies via volume maximizing lunes. Finally, we give estimates for partial coverings of balls by planks in Euclidean d -space.

8.1 Proof of Theorem 7.1.8 Theorem 7.1.8. Let K and C be convex bodies in Ed , d 2 and let the positive integer m be given. If K is sliced into n 1 pieces by n 1 successive hyperplane cuts (i.e., when each cut divides one piece), then the minimum of the greatest mth successive C-inradius of the pieces is the .mn/th successive C-inradius of K (i.e., rC .K; mn/). An optimal partition is achieved by n 1 parallel hyperplane cuts equally spaced along the minimal C-width of the rC .K; mn/C-rounded body of K.

8.1.1 On Coverings of Convex Bodies by Two Planks On the one hand, the following statement is an extension to higher dimensions of Theorem 4 in [3]. On the other hand, the proof presented below is based on Theorem 4 of [3]. Lemma 8.1.1. If a convex body K in Ed ; d 2 is covered by the planks P1 and P2 , then wC .P1 / C wC .P2 / wC .K/ for any convex body C in Ed . Proof. Let H1 (resp., H2 ) be one of the two hyperplanes which bound the plank P1 (resp., P2 ). If H1 and H2 are translates of each other, then the claim is K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 8, © Springer International Publishing Switzerland 2013

143

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8 Proofs on Coverings by Cylinders

obviously true. Thus, without loss of generality we may assume that L WD H1 \ H2 is a .d 2/-dimensional affine subspace of Ed . Let E2 be the 2-dimensional linear subspace of Ed that is orthogonal to L. If ./0 denotes the (orthogonal) projection of Ed parallel to L onto E2 , then obviously, wC0 .P01 / D wC .P1 /, wC0 .P02 / D wC .P2 / and wC0 .K0 / wC .K/. Thus, it is sufficient to prove that wC0 .P01 / C wC0 .P02 / wC0 .K0 /: In other words, it is sufficient to prove Lemma 8.1.1 for d D 2. Hence, in the rest of the proof, K; C; P1 ; P2 ; H1 , and H2 mean the sets introduced and defined above, however, for d D 2. Now, we can make the following easy observation wC .P1 / C wC .P2 / D D

w.P2 / w.P1 / C w.C; H1 / w.C; H2 /

w.P2 / w.K; H2 / w.P1 / w.K; H1 / C w.K; H1 / w.C; H1 / w.K; H2 / w.C; H2 / w.P2 / w.P1 / wC .K/ C w.K; H1 / w.K; H2 / D .wK .P1 / C wK .P2 // wC .K/:

Then recall that Theorem 4 in [3] states that if a convex set in the plane is covered by planks, then the sum of their relative widths is at least 1. Thus, using our terminology, we have that wK .P1 / C wK .P2 / 1, finishing the proof of Lemma 8.1.1.

8.1.2 Minimizing the Greatest mth Successive C-Inradius Let K and C be convex bodies in Ed , d 2. We prove Theorem 7.1.8 by induction on n. It is trivial to check the claim for n D 1. So, let n 2 be given and assume that Theorem 7.1.8 holds for at most n 2 successive hyperplane cuts and based on that we show that it holds for n 1 successive hyperplane cuts as well. The details are as follows. Let H1 ; : : : ; Hn1 denote the hyperplanes of the n 1 successive hyperplane cuts that slice K into n pieces such that the greatest mth successive C-inradius of the pieces is the smallest possible say, . Then take the first cut H1 that slices K into the pieces K1 and K2 such that K1 (resp., K2 ) is sliced into n1 (resp., n2 ) pieces by the successive hyperplane cuts H2 ; : : : ; Hn1 , where n D n1 C n2 . The induction hypothesis implies that rC .K1 ; mn1 / DW 1 and rC .K2 ; mn2 / DW 2 and therefore wC .K1 C / wC .K1 1 C / D mn1 1 mn1 I moreover,

(8.1)

8.2 Proof of Theorem 7.1.12

145

wC .K2 C / wC .K2 2 C / D mn2 2 mn2 :

(8.2)

Now, we need to define the following set. Definition 8.1.2. Assume that the origin o of Ed belongs to the interior of the convex body C Ed . Consider all translates of C which are contained in the convex body K Ed . The set of points in the translates of C that correspond to o form a convex set called the inner C-parallel body of K denoted by K C . Clearly, .K1 / C [ .K2 / C K C with .K1 / C \ .K2 / C D ;: Also, it is easy to see that there is a plank P with wC .P/ D such that it is parallel to H1 and contains H1 in its interior; moreover, K C .K1 / C [ .K2 / C [ P: Hence, applying Lemma 8.1.1 to .K1 / C as well as .K2 / C we get that wC K C wC .K1 / C C C wC .K2 / C : (8.3) D wC .K1 C / , It follows from the definitions that wC .K1 / C

C

C wC .K2 / C D wC .K2 / and wC K C D wC .K / . Hence, (8.3) is equivalent to wC .K C / wC .K1 C / C wC .K2 C /:

(8.4)

Finally, (8.1),(8.2), and (8.4) yield that wC .K C / mn1 C mn2 D mn :

(8.5)

Thus, (8.5) clearly implies that rC .K; mn/ . As the case, when the optimal partition is achieved, follows directly from the definition of the mnth successive C-inradius of K, the proof of Theorem 7.1.8 is complete.

8.2 Proof of Theorem 7.1.12 Theorem 7.1.12. Let K and C be convex bodies in Ed ; d 2 and let m be a positive integer. If V1 [ V2 [ [ Vn is an inductive partition of Ed such that int.Vi \ K/ ¤ ; for all 1 i n, then n X i D1

rC .Vi \ K; m/ rC .K; m/:

146

8 Proofs on Coverings by Cylinders

8.2.1 On an Extension of a Helly-Type Result of Klee Recall the following Helly-type result of Klee [137]. Let F WD fAi j i 2 I g be a family of compact convex sets in Ed ; d 2 containing at least d C 1 members. Suppose C is a compact convex set in Ed such that the following holds: For each subfamily of d C 1 sets in F , there exists a translate of C that is contained in all d C 1 of them. Then there exists a translate of C that is contained in all the members of F . In what follows we give a proof of the following extension of Klee’s theorem to linear packings. Theorem 8.2.1. Let F WD fAi j i 2 I g be a family of convex bodies in Ed ; d 2 containing at least d C 1 members. Suppose C is a convex body in Ed and m 1 is a positive integer moreover, l is a line passing through the origin o in Ed such that the following holds: For each subfamily of d C 1 convex bodies in F , there exists a linear packing of m translates of C with separating direction l that is contained in all d C 1 of them. Then there exists a linear packing of m translates of C with separating direction l that is contained in all the members of F . Proof. Let C1 ; : : : ; Cm be a linear packing of m translates of C with separating direction l in Ed . In what follows we always assume that whenever we write C1 ; : : : ; Cm , then the orthogonal projections Prl .C1 /; : : : ; Prl .Cm / of the translates C1 ; : : : ; Cm of C onto the line l, listed in this order, form a consecutive sequence of congruent non-overlapping intervals with respect to some fixed orientation of l. Let Li denote the family of all linear packings of m translates of C with separating direction l which lie in Ai . Moreover, without loss of generality we may assume that the origin o of Ed is in C. Now, for each Ai , i 2 I let .Ai /C WD ft 2 Ed j t C C is the first member of a linear packing in Li g: Clearly, each .Ai /C , i 2 I is a compact set. Furthermore, we claim that .Ai /C is a convex set for all i 2 I . Indeed, it is rather straightforward to show that if t1 C C; C2 ; : : : ; Cm1 ; Cm ; and t01 C C; C02 ; : : : ; C0m1 ; C0m are linear packings chosen from Li , then there exists a linear packing t1 C .1 /t01 C C; C002 ; : : : ; C00m1 ; Cm C .1 /C0m of m translates of C with separating direction l contained in the convex hull conv . t1 C .1 /t01 C C/ [ . Cm C .1 /C0m / such that it is also in Li for any given 0 1. Thus, based on the assumptions of Theorem 8.2.1, the family f.Ai /C j i 2 I g is a collection of compact convex sets of Ed with the property that

8.2 Proof of Theorem 7.1.12

147

.Ai1 /C \ .Ai2 /C \ \ .Aid C1 /C ¤ ; for all pairwise distinct i1 ; i2 ; : : : ; id C1 2 I . Hence, Helly’s theorem (see for example [17], p. 19) implies that \f.Ai /C j i 2 I g ¤ ;, finishing the proof of Theorem 8.2.1.

8.2.2 On Some Concave Functions of Successive Inradii A rather straightforward extension of the method of Akopyan and Karasev [2] combined with Theorem 8.2.1 gives the following statement. For the statement below as well as its proof we extend the definition of the mth successive C-inradius of convex bodies K Ed via including all non-empty compact convex sets K Ed having intK D ; with the definition rC .K; m/ WD 0 and via including the empty set ; with the definition rC .;; m/ WD 1. Theorem 8.2.2. Let K and C be convex bodies in Ed ; d 2 and let m be a positive integer. Moreover, let V1 [ V2 [ [ Vn be an inductive partition of Ed and let Ki .x/ WD K \ .x C Vi / for all x 2 Ed and 1 i n. Then the function r.x; m/ WD

n X

rC .Ki .x/; m/

i D1

is a concave function of x 2 Ed . Proof. First, we need the following statement. Lemma 8.2.3. Let P WD fx 2 Ed j Li .x/ 0 for al l 1 i pg be an arbitrary d -dimensional convex polytope of Ed defined by the linear inequalities fLi .x/ 0; 1 i pg. If P.y/ WD fx 2 Ed j Li .x/ C yi 0 for al l 1 i pg stands for the parallel deformation of P generated by the vector y D .y1 ; : : : ; yp / 2 Ep , then rC .P.y/; m/ is a concave function of y for any fixed convex body C of Ed and for any fixed positive integer m. Proof. Let l be an arbitrary line passing through the origin o in Ed . Moreover, let rC .P.y/; m j l/ denote the largest real > 0 with the property that there exists a linear packing of m translates of C lying in P.y/ and having l as a separating direction. Furthermore, let PI .y/ WD fx 2 Ed j Li .x/ C yi 0 for all i 2 I g

148

8 Proofs on Coverings by Cylinders

whenever I Œp WD f1; 2; : : : ; pg and let rC .PI .y/; m j l/ be defined similarly to rC .P.y/; m j l/. The definition of successive inradii based on linear packings and Theorem 8.2.1 imply in a straightforward way that rC .P.y/; m/ D

rC .P.y/; m j l/ D

inf rC .P.y/; m j l/

o2lEd

inf

I Œp; card.I /d C1

rC .PI .y/; m j l/

(8.6)

(8.7)

Next, we take a closer look of the convex polyhedral set PI .y/: Clearly, PI .y/ is either a (convex polyhedral) cylinder (of some convex polyhedral base set having dimension strictly less than d ), or a (convex polyhedral) cone, or a d -dimensional simplex. In the first case, we use induction on the dimension of the base set. In the second case, we always have that rC .PI .y/; m j l/ D C1. In the third case, it is easy to see (using the definition of successive inradii based on linear packings) that rC .PI .y/; m j l/ is a linear function of y. Thus, as the infimum of concave functions is concave, (8.6) and (8.7) imply in a straightforward way that rC .P.y/; m/ is a concave function of y, finishing the proof of Lemma 8.2.3. Second, observe that Lemma 8.2.3 and standard approximation by polytopes yield the following statement. Corollary 8.2.4. Let K1 ; : : : ; KN , and C be convex bodies in Ed and let m be a positive integer. Then rC ..y1 C K1 / \ \ .yN C KN /; m/ is a concave function of .y1 ; : : : ; yN / 2 ENd . Finally, using Corollary 8.2.4 we get that rC .Ki .x/; m/ P is a concave function of x 2 Ed for all 1 i n and therefore also r.x; m/ D niD1 rC .Ki .x/; m/ is a concave function of x 2 Ed , finishing the proof of Theorem 8.2.2.

8.2.3 Estimating Sums of Successive Inradii Now, we are set for an inductive proof of Theorem 7.1.12 on the number n of tiles in the relevant inductive P partition. The details are as follows. By Theorem 8.2.2 the function r.x; m/ D niD1 rC .Ki .x/; m/ is a concave function of x and so, Xr WD fx 2 Ed j r.x; m/ > 1g is a closed convex set in Ed . If x0 is a boundary point of Xr , then at least one Ki .x0 / D K \ .x0 C Vi / must have an empty interior in Ed say, intKi0 .x0 / D ; for some 1 i0 n. Then take the inductive partition d W1 [ [Wi0 1 [Wi0 C1 [ [Wn of E such that Vj Wj for all j ¤ i0 . Now, it is easy to see that if int K \ .x0 C Vj / ¤ ; for some j ¤ i0 , then K\.x0 CVj / D

8.3 Proof of Theorem 7.2.2

149

K \ .x0 C Wj /. Thus, n X

rC .K \ .x0 C Vi /; m/ D

i D1

X

rC K \ .x0 C Wj /; m

j ¤i0

and therefore by induction we get that the inequality r.x0 ; m/ rC .K; m/ holds for all boundary points x0 of Xr . Then this fact and the concavity of r.x; m/ imply in a straightforward way that the inequality r.x; m/ rC .K; m/ holds for all x 2 Xr unless Xr is a closed halfspace of Ed . However, the latter case can happen only when (each Vi , 1 i n contains the same halfspace and therefore) n D 1. As Theorem 7.1.12 clearly holds for n D 1, our inductive proof of Theorem 7.1.12 is complete.

8.3 Proof of Theorem 7.2.2 Theorem 7.2.2. Let K be a convex body in S Ed . Let C1 ; : : : ; CN be k-codimensional d cylinders in E ; 0 < k < d such that K N i D1 Ci : Then N X i D1

1 crvK .Ci / d : k

Moreover, if K S is an ellipsoid and C1 ; : : : ; CN are 1-codimensional cylinders in Ed such that K N i D1 Ci , then N X

crvK .Ci / 1:

i D1

8.3.1 Covering Ellipsoids by 1-Codimensional Cylinders We prove the part of Theorem 7.2.2 on ellipsoids. In this case, every Ci can be presented as Ci D li C Bi , where li is a line containing the origin o in Ed and Bi is a measurable set in Ei WD li? . Because crvK .C/ D crvT K .T C/ for every invertible affine map T W Ed ! Ed , we therefore may assume that K D Bd , where Bd denotes the unit ball centered at the origin o in Ed . Recall that !d 1 WD vold 1 .Bd 1 /. Then crvK .Ci / D

vold 1 .Bi / : !d 1

Consider the following (density) function on Ed ,

150

8 Proofs on Coverings by Cylinders

p p.x/ WD 1= 1 kxk2 for kxk < 1 and p.x/ WD 0 otherwise, where k k denotes the standard Euclidean norm in Ed . The corresponding measure on Ed we denote by , that is, d.x/ D p.x/d x. Let l be a line containing o in Ed and E D l ? . It follows from direct calculations that for every z 2 E with kzk < 1, Z p.x/ d x D : lCz

Thus, we have Z

Z

Z

.B / D

p.x/ d x D

d

p.x/ d x d z D !d 1

Bd \E

Bd

lCz

and for every i N , Z p.x/ d x D Bi

Ci

Because Bd

SN

!d 1

Z

Z

.Ci / D

i D1 Ci ,

li Cz

p.x/ d x d z D vold 1 .Bi /:

we obtain

X [ N N N X D .B / Ci .Ci / D vold 1 .Bi /: d

i D1

i D1

i D1

It implies N X

crvBd .Ci / D

i D1

N X vold 1 .Bi / i D1

!d 1

1:

8.3.2 Covering Convex Bodies by Cylinders N i WD We show the general case of Theorem 7.2.2 as follows. For i N denote C ? Ci \ K and Ei WD Hi and note that K

N [

Ni C

and

N i D Bi \ PEi K: PEi C

i D1

N i K we also have Because C N i \ .x C Hi // max volk .K \ .x C Hi //: max volk .C x2Ed

x2Ed

We use the following theorem, proved by Rogers and Shephard [161] (see also [75] and Lemma 8.8 in [157]).

8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes

151

Theorem 8.3.1. Let 1 k d 1. Let K be a convex body in Ed and E be a k-dimensional linear subspace of Ed . Then ! d vold .K/: max vold k .K \ .x C E // volk .PE K/ k x2Ed ?

We note that the reverse estimate max vold k .K \ .x C E ? // volk .PE K/ vold .K/

x2Ed

is a simple application of the Fubini theorem and is correct for any measurable set K in Ed . Thus, applying Theorem 8.3.1 (and the remark after it, saying that we don’t need N i ) we obtain for every i N : convexity of C crvK .Ci / D

maxx2Ed

N i/ vold k .PEi C vold k .Bi / vold k .PEi K/ vold k .PEi K/

N i/ vold .C maxx2Ed volk .K \ .x C Hi // d N volk .Ci \ .x C Hi // vold .K/ k N i/ vold .C d : vold .K/ k

N i s cover K, we observe Using that C N X i D1

1 crvK .Ci / d ; k

which completes the proof of Theorem 7.2.2.

8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes Theorem 7.3.4. If K is a spherically convex body in Sd ; d 2, then Svold .K/

.d C 1/!d C1 r.K/:

Equality holds if and only if K is a lune. On the one hand, the proof that follows is essentially the same as the original one published in [55]. On the other hand, it is yet somewhat more simple and its organization follows the relevant discussion in [2].

152

8 Proofs on Coverings by Cylinders

Lemma 8.4.1. Let be a spherically symmetric and absolute continuous (finite) measure on Ed . Moreover, let B be a closed ball centered at the origin o in Ed , and let T be a closed starshaped set with center o in Ed . Then B \ T Ed .B/.T/ : Proof. Let us decompose Ed into finitely many convex cones Vi ofequal measures .Vi / having o as an apex in common in Ed . Here the measures B \ Vi are all equal due to the spherical symmetry of . (In other words, we start with a simple needle decomposition of Ed . For more details on needle decompositions see for example, [152].) Thus, Lemma 8.4.1 will follow from the inequality B \ T \ Vi Vi .B \ Vi /.T \ Vi /

(8.8)

by summation. As the needle decomposition at hand can be made so that every Vi gets arbitrarily close to a halfline starting at o in Ed therefore the absolute continuity of implies that the limit case of (8.8) becomes the following inequality on nonnegative functions: Z 0

minfx;yg

Z

Z

C1

f .t/dt

f .t/dt 0

Z

x

y

f .t/dt 0

f .t/dt : 0

This latter inequality follows from the trivial observation that minfx; yg z xy for any x; y 2 Œ0; z. Lemma 8.4.2. Let H be a closed hemisphere with center o in Sd and let B be a closed ball of spherical radius at most 2 and of center o in Sd . If T is a closed starshaped set with center o in H, then Svold .B \ T/Svold .H/ Svold .B/Svold .T/ for the standard measure (i.e., spherical Lebesgue measure) Svold ./ on Sd . Proof. Let Ed be the hyperplane tangent to the unit sphere Sd at the point o in Ed C1 . Then the central projection of Sd onto Ed (keeping the point o fixed) combined with Lemma 8.4.1 implies Lemma 8.4.2. For the purpose of the next statement let BŒx; " denote the closed ball of spherical radius " 2 centered at the point x in Sd . Furthermore, for any subset X of Sd let X" WD [x2X BŒx; " be called the "-neighbourhood of X in Sd . Lemma 8.4.3. Let X be a closed subset of Sd not lying on an open hemisphere of Sd . Then for any " 2 the following inequality holds: O "/ ; Svold .X" / Svold .X O is a pair of antipodal points of Sd . where X

8.4 Proof of Theorem 7.3.4: Volume Maximizing Lunes

153

Fig. 8.1 The point xi and its Voronoi cell Vi and the hemisphere Hi centered at xi

Proof. It is sufficient to prove Lemma 8.4.3 for finite X say, for X WD fx1 ; x2 ; : : : ; xm g Sd . Take the (nearest point) Voronoi tiling of Sd generated by X with Vi standing for the Voronoi cell assigned to the point xi , 1 i m. Let Hi be the closed hemisphere of Sd centered at xi , 1 i m. (See Fig. 8.1.) As, by assumption, X does not lie on an open hemisphere of Sd therefore, Vi Hi holds for all 1 i m. Thus, Lemma 8.4.2 yields that Svold .BŒxi ; " \ Vi /Svold .Hi / Svold .BŒxi ; "/Svold .Vi / ; or equivalently Svold .Sd / Svold .BŒxi ; "/Svold .Vi / (8.9) 2 Pm holds for all 1 i m. As i D1 Svold .X" \ Vi / D Svold .X" / and P m d i D1 Svold .Vi / D Svold .S / therefore (8.9) implies in a straightforward way that Svold .X" \ Vi /

Svold .X" / 2Svold .BŒx; "/ holds for any x 2 Sd , finishing the proof of Lemma 8.4.3.

Lemma 8.4.4. Let be a spherically symmetric and absolute continuous (finite) measure on Ed . Moreover, let K be a convex body in Ed with the inscribed (closed) ball B centered at the origin o in Ed . Then .K/ .P/ ; where P is a plank of Ed having B as an inscribed ball.

154

8 Proofs on Coverings by Cylinders

Proof. As the measure is absolute continuous therefore representing it as an integral, it is sufficient to prove the inequality .K \ S / .P \ S /

(8.10)

for any .d 1/-dimensional sphere S of Ed with center o and with the proper spherical Lebesgue measure on it. Let r (resp., r) be the radius of S (resp., B). If r r, then the inequality (8.10) is obvious. So, we are left with the case r < r. The more exact details are as follows. Let X 0 WD bd K \ B (resp., XO 0 WD bd P \ B) and X WD rr X 0 S (resp., XO WD rr XO 0 S ). Clearly, X is not contained in an open hemisphere of S and of course, XO is a pair of antipodal points on S . Now, let " WD arccos rr < 2 and for any Y S let Y" denote the set of those points of S whose angular distance from Y is at most " in S . It is straightforward to see that X" \ .int K/ \ S D ; (8.11) and XO" \ .int P/ \ S D ; and XO " [ P \ S D S :

(8.12)

.X" / .XO" / :

(8.13)

By Lemma 8.4.3

Thus, (8.11)–(8.13) imply in a straightforward way that .K \ S / .S / .X" / .S / .XO" / D .P \ S / ; finishing the proof of (8.10) as well as Lemma 8.4.4.

d C1

Finally, let E be the hyperplane of E that is tangent to the unit sphere Sd touching it at the center of the inscribed ball of the convex body K in Sd . Then the central projection of Sd (from its center) onto Ed combined with Lemma 8.4.4 proves Theorem 7.3.4. As a last note we mention that the above proof implies the following stronger (truncated) version of Theorem 7.3.4. d

Theorem 8.4.5. Let K be a convex body in Sd with the inscribed (closed) ball B centered at o 2 Sd . Furthermore, let B0 be any (closed) ball of Sd centered at o. Then Svold K \ B0 Svold L \ B0 ; where L is a lune of Sd having B as an inscribed ball.

8.6 Proof of Theorem 7.4.6: Lower Bounds for Ball-Polyhedra

155

8.5 Proof of Theorem 7.4.2: Partial Coverings in Euclidean 3-Space Theorem 7.4.2. Let w1 ; w2 ; : : : ; wn be positive real numbers satisfying the inequality w1 C w2 C C wn < 2. Then the union of the planks P1 ; P2 ; : : : ; Pn of width w1 ; w2 ; : : : ; wn in E3 covers the largest volume subset of B3 if and only if P1 [ P2 [ [ Pn is a plank of width w1 C w2 C C wn with o as a center of symmetry. Let P1 ; P2 ; : : : ; Pn be an arbitrary family of planks of width w1 ; w2 ; : : : ; wn in E3 and let P be a plank of width w1 C w2 C C wn with o as a center of symmetry. Moreover, let S.x/ denote the sphere of radius x centered at o. Now, recall the wellknown fact that if P.y/ is a plank of width y whose both boundary planes intersect S.x/, then sarea.S.x/ \ P.y// D 2xy, where sarea. : / refers to the surface area measure on S.x/. This implies in a straightforward way that sareaŒ.P1 [ P2 [ [ Pn / \ S.x/ sarea.P \ S.x//; and so, Z

1

vol3 ..P1 [ P2 [ [ Pn / \ B3 / D

sareaŒ.P1 [ P2 [ [ Pn / \ S.x/ dx

0

Z

1

sarea.P \ S.x// dx D vol3 .P \ B3 /;

0

finishing the proof of the “if” part of Theorem 7.4.2. Actually, a closer look of the above argument gives a proof of the “only if” part as well.

8.6 Proof of Theorem 7.4.6: Lower Bounds for Ball-Polyhedra Theorem 7.4.6. Let BŒX Ed be a ball-polyhedron of minimal width x with 1 x < 2. Then 2x;d 2x;d / C svold 1 .bd.KBL //.x 1/ C vold .Bd /.x 1/d . vold .BŒX / vold .KBL Recall that if X is a finite set lying in the interior of a unit ball in Ed , then we can talk about its spindle convex hull convs .X /, which is simply the intersection of all (closed) unit balls of Ed that contain X (for more details see [59]). The following statement can be obtained by combining Corollary 3.4 of [59] and Proposition 1 of [58].

156

8 Proofs on Coverings by Cylinders

Lemma 8.6.1. Let X be a finite set lying in the interior of a unit ball in Ed . Then

(i) convs .X / D B BŒX and therefore BŒX D B convs .X / , (ii) The Minkowski sum BŒX C conv s .X / is a convex body of constant width 2 in Ed and so, w.BŒX / C diam convs .X / D 2, where diam. : / stands for the diameter of the corresponding set in Ed . By part .ii/ of Lemma 8.6.1, diam convs .X / 2 x. This implies, via a classical theorem of convexity (see, e.g., [65]), the existence of a convex body L of constant width .2 x/ in Ed with conv s .X / L.

Hence, using part .i / of Lemma 8.6.1, we get that BŒL BŒX D B convs .X / . Finally, notice that as L is a convex body of constant width .2 x/ therefore BŒL is, in fact, the outer-parallel domain of L having radius .x 1/ (i.e., BŒL is the union of all d -dimensional (closed) balls of radii .x 1/ in Ed that are centered at the points of L). Thus, vold .BŒX / vold .BŒL/ D vold .L/ C svold 1 .bd.L//.x 1/ C vold .Bd /.x 1/d : The above inequality together with the following obvious ones 2x;d

2x;d vold .L/ vold .KBL / and svold 1 .bd.L// svold 1 .bd.KBL

implies Theorem 7.4.6 in a straightforward way.

//

Chapter 9

Research Problems: An Overview

Abstract In this chapter we give a short summary of the 30 open research problems that have been raised in this book. Some of them are well known however, the majority of them seems to be new. For more details we refer the interested reader to the relevant sections of this book.

9.1 Unit Sphere Packings 9.1.1 The Contact Number Problem of Finite Sphere Packings For a given positive integer n 2 let C.n/ denote the largest number of touching pairs in a packing of n congruent balls in E3 . Problem 1.1.4. Find those positive integers n for which C.n/ can be achieved in a packing of n unit balls in E3 consisting of parallel layers of unit balls each being a subset of the densest infinite hexagonal layer of unit balls. Conjecture 1.1.5. C.12/ D 33 and C.13/ D 36. Problem 1.1.8. Does the limit limn!C1

6nC.n/ 2

exist?

n3

9.1.2 Dense Sphere Packings in Euclidean 3-Space Let T be a tiling of the 3-dimensional Euclidean Pi ; i D 1; 2; : : : each containing a unit ball 3-dimensional ball Bi having radius 1 for i D there is a finite upper bound for the diameters

space E3 into convex polyhedra say, Pi containing the closed 1; 2; : : : . Also, we assume that of the convex cells in T , i.e.,

K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8 9, © Springer International Publishing Switzerland 2013

157

158

9 Research Problems: An Overview

supfdiam.Pi /ji D 1; 2; : : : g < 1, where diam./ denotes the diameter of the corresponding set. In short, we say that T is a normal tiling of E3 with the underlying packing P of the unit balls Bi ; i D 1; 2; : : : . Then we define the (lower) average surface area s.T / of the cells in T as follows: P s.T / WD lim inf L!1

fi jBi CL g

sarea.Pi \ CL /

cardfi jBi CL g

;

where CL denotes the cube centered at the origin o with edges parallel to the coordinate axes of E3 and having edge length L furthermore, sarea./ and card./ denote the surface area and cardinality of the corresponding sets. (We note that it is rather straightforward to show that s.T / is independent from the choice of the coordinate system of E3 .) Problem 1.3.1. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? Conjecture 1.3.4. If Tpdenotes the Voronoi tiling of an arbitrary unit ball packing in E3 , then s.T / 12 2 D 16:9705 : : :. Problem 1.3.6. If the Euclidean3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average edge curvature of the cells?

9.1.3 On Sphere Packings in High Dimensions Conjecture 1.4.5. The E8 root lattice (resp., the Leech lattice) represents the largest possible density for unit sphere packings in E8 (resp., E24 ). Conjecture 1.4.8. Let 5 d 41 be given. Then the volume of the convex hull of n 1 non-overlapping unit balls in Ed is at least as large as the volume of the convex hull of the “sausage arrangement” of n non-overlapping unit balls which is an arrangement whose centers lie on a line of Ed such that the unit balls of any two consecutive centers touch each other.

9.1.4 Uniformly Stable and Periodic Extreme Lattices A circle packing in the plane of constant curvature is called solid if no finite subset of the circles can be rearranged such that the rearranged circles together with the rest of the circles form a packing not congruent to the original.

9.2 Contractions of Sphere Arrangements

159

Conjecture 1.5.2. The incircles of a regular tiling fp; 3g form a strongly solid packing for p D 6 as well as for p D 7, that is, by removing any circle from the packing the remaining circles still form a solid packing. Conjecture 1.5.5. The largest density of unit ball packings in Ed ; d 4 can always be attained by periodic packings of unit balls.

9.2 Contractions of Sphere Arrangements 9.2.1 The Kneser–Poulsen Conjecture Recall that k k denotes the standard Euclidean norm of the d -dimensional Euclidean space Ed . So, if pi ; pj are two points in Ed , then kpi pj k denotes the Euclidean distance between them. It is convenient to denote the (finite) point configuration consisting of the points p1 ; p2 ; : : : ; pN in Ed by p D .p1 ; p2 ; : : : ; pN /. Now, if p D .p1 ; p2 ; : : : ; pN / and q D .q1 ; q2 ; : : : ; qN / are two configurations of N points in Ed such that for all 1 i < j N the inequality kqi qj k kpi pj k holds, then we say that q is a contraction of p. Finally, let Bd Œpi ; ri denote the (closed) d -dimensional ball centered at pi with radius ri in Ed and let vold ./ represent the d -dimensional volume (Lebesgue measure) in Ed . Conjecture 3.1.1. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed ; d 3, then vold

N [

! B Œpi ; ri vold d

i D1

N [

! d

B Œqi ; ri :

i D1

Conjecture 3.1.2. If q D .q1 ; q2 ; : : : ; qN / is a contraction of p D .p1 ; p2 ; : : : ; pN / in Ed ; d 3, then vold

N \ i D1

! Bd Œpi ; ri vold

N \

! Bd Œqi ; ri :

i D1

9.2.2 Kneser–Poulsen-Type Theorems in Non-Euclidean Spaces Problem 3.4.3. Suppose that p D .p1 ; p2 ; : : : ; pN / and q D .q1 ; q2 ; : : : ; qN / are two point configurations in S2 . Then prove or disprove that there is a monotone piecewise-analytic motion from p D .p1 ; p2 ; : : : ; pN / to q D .q1 ; q2 ; : : : ; qN / in S4 .

160

9 Research Problems: An Overview

Problem 3.4.4. Let P and Q be compact convex polyhedra of H3 with P (resp., C C C Q) being the intersection of the closed halfspaces HP;1 ; HP;2 ; : : : ; HP;N (resp., C C C C C HQ;1 ; HQ;2 ; : : : ; HQ;N ). Assume that the dihedral angle HQ;i \ HQ;j (containing C C \ HP;j (containing Q) is at least as large as the corresponding dihedral angle HP;i P) for all 1 i < j N . Then prove or disprove that the volume of P is at least as large as the volume of Q.

9.2.3 Alexander’s Conjecture Revisited Conjecture 3.6.1. Under arbitrary contraction of the center points of finitely many congruent disks in the Euclidean plane, the perimeter of the intersection of the disks cannot decrease.

9.3 Ball-Polyhedra and Spindle Convex Bodies Based on our interests in discrete geometry we pay special attention to intersections with non-empty interiors of finitely many congruent closed d -dimensional balls in Ed . In fact, one may assume that the congruent d -dimensional balls in question are of unit radius; that is, they are unit balls of Ed . Also, it is natural to assume that removing any of the unit balls defining the intersection in question yields the intersection of the remaining unit balls becoming a larger set. In other words, whenever we take an intersection of finitely many unit balls we always assume that it is generated by a reduced family of unit balls. If d D 2, then we call the sets in question disk-polygons and for d 3 they are called ball-polyhedra. In general, due to the underlying convexity, intersections with non-empty interiors of not necessarily finitely many unit balls in Ed are called spindle convex bodies.

9.3.1 Disk-Polygons Revisited Take a disk-polygon D in E2 . Then choose a positive " not larger than the inradius of D (which is the radius of the largest circular disk contained in D) and take the union of all circular disks of radius " that lie in D. We call the set obtained in this way the "-rounded disk-polygon of D and denote it by D."/. Conjecture 5.2.4. Let D be a disk-polygon in E2 with the property that the pairwise distances between the centers of its generating unit disks are at most 1. Then any of the shortest (generalized) billiard trajectories in the "-rounded disk-polygon D."/ is a 2-periodic one for all " being at most as large as the inradius of D.

9.3 Ball-Polyhedra and Spindle Convex Bodies

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Problem 5.2.6. Let P be a ball-polyhedron in Ed ; d 3 with the property that the pairwise distances between the centers of its generating unit balls are at most 1. Then prove or disprove that any of the shortest generalized billiard trajectories in P is a 2-periodic one. p The disk-polygon D is called a disk-polygon with center parameter t, 0 < t < 3 D 1:732 : : : , if the distance between any two centers of the generating unit disks of D is at most t. Let F .t/ denote the family of all disk-polygons with center parameter t. Let .t/ denote the regular disk-triangle whose three generating unit disks are centered at the vertices of a regular triangle of side length t, 1 t < p 3 D 1:732 : : : . Problem 5.2.8. p Let D 2 F .t/ be an arbitrary disk-polygon with center parameter t; 1 t < 3. Prove or disprove that the perimeter of D is at least as large as the perimeter of .t/; that is, p.D/ p..t//:

9.3.2 On a Steinitz-Type Problem for Ball-Polyhedra One can represent the boundary of a ball-polyhedron in E3 as the union of vertices, edges, and faces defined in a rather natural way as follows. A boundary point is called a vertex if it belongs to at least three of the closed unit balls defining the ball-polyhedron. A face of the ball-polyhedron is the intersection of one of the generating closed unit balls with the boundary of the ball-polyhedron. Finally, if the intersection of two faces is non-empty, then it is the union of (possibly degenerate) circular arcs. The non-degenerate arcs are called edges of the ball-polyhedron. Obviously, if a ball-polyhedron in E3 is generated by at least three unit balls, then it possesses vertices, edges, and faces. Clearly, the vertices, edges and faces of a ball-polyhedron (including the empty set and the ball-polyhedron itself) are partially ordered by inclusion forming the vertex-edge-face structure of the given ball-polyhedron. A ball-polyhedron in E3 is called a standard ball-polyhedron if its vertex-edge-face structure is a lattice (with respect to containment). This is the case if, and only if, the intersection of any two faces is either empty, or one vertex or one edge, and every two edges share at most one vertex. In this case, we simply call the vertex-edge-face structure in question the face lattice of the standard ballpolyhedron. Problem 5.3.2. Prove or disprove that every simple, planar, and 3-connected graph is the edge-graph of some standard ball-polyhedron in E3 .

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9.3.3 On Global and Local Rigidity of Ball-Polyhedra We say that the standard ball-polyhedron P in E3 is globally rigid with respect to its face angles (resp., globally rigid with respect to its inner dihedral angles) within the family of standard ball-polyhedra if the following holds. If P0 is another standard ball-polyhedron in E3 whose face lattice is combinatorially equivalent to that of P and whose face angles (resp., inner dihedral angles) are equal to the corresponding face angles (resp., inner dihedral angles) of P, then P0 is congruent to P. Problem 5.4.5. Prove or disprove that every standard ball-polyhedron of E3 is globally rigid with respect to its face angles within the family of standard ballpolyhedra. Problem 5.4.8. Prove or disprove that every standard ball-polyhedron of E3 is globally rigid with respect to its inner dihedral angles within the family of standard ball-polyhedra. Recall that a ball-polyhedron is called a simple ball-polyhedron, if at every vertex exactly three edges meet. Conjecture 5.4.9. Let P be a simple and standard ball-polyhedron of E3 . Then P is globally rigid with respect to its inner dihedral angles within the family of standard ball-polyhedra.

9.3.4 Illuminating Ball-Polyhedra and Spindle Convex Bodies By a convex body C in Sd 1 we understand the intersection Sd 1 \ Co , where Co stands for a line-free d -dimensional closed convex cone with apex o in Ed . We denote by KSd 1 the space of all convex bodies in Sd 1 , equipped with the Hausdorff metric. L Sd 1 is called a lune of Sd 1 if it is the intersection of two (distinct) closed hemispheres of Sd 1 having nonempty interior. The width of L is simply the angular measure of the dihedral angle pos.L/, where pos./ refers to the positive hull of the corresponding set in Ed . The minimal width Swidth.C/ of C 2 KSd 1 is the smallest width of the lunes that contain C. Also, we say that C 2 KSd 1 is a convex body of constant width w if w D Swidth.C/ D Sdiam.C/, where Sdiam./ refers to the spherical diameter of the corresponding set in Sd 1 . Conjecture 5.7.6. Among all convex bodies of constant width 2 in Sd 1 , d 4, the .d 1/-dimensional regular simplex of edge length 2 has the smallest volume.

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9.4 Coverings by Cylinders 9.4.1 Plank Theorems: Old and New Let C and K be convex bodies in Ed and let H be a hyperplane of Ed . Then the distance w.C; H / between the two supporting hyperplanes of C parallel to H is called the width of C parallel to H . Moreover, the C-width of K parallel to H is / denoted by wC .K; H / and is defined as w.K;H w.C;H / . The minimal C-width of K is denoted by wC .K/ and is defined as the minimum of wC .K; H /, where the minimum is taken over all possible hyperplanes H of Ed . Recall that the inradius of K is the radius of the largest ball contained in K. It is quite natural then to introduce the C-inradius of K as the factor of the largest positive homothetic copy of C, a translate of which is contained in K. We need to do one more step to introduce the so-called successive C-inradii of K as follows. Let r be the C-inradius of K. For any 0 < r let the C-rounded body of K be denoted by K C and be defined as the union of all translates of C that are covered by K. Now, take a fixed integer m 1. On the one hand, if > 0 is sufficiently small, then wC .K C / > m . On the other hand, wC .KrC / D r mr. As wC .K C / is a decreasing continuous function of > 0 and m is a strictly increasing continuous function of , there exists a uniquely determined > 0 such that wC .K C / D m : This uniquely determined is called the mth successive C-inradius of K and is denoted by rC .K; m/. Conjecture 7.1.9. Let K and C be convex bodies in Ed ; d 2 and let the positive integer m be given. If K is sliced by n 1 hyperplanes, then the minimum of the greatest mth successive C-inradius of the pieces is the .mn/th successive C-inradius of K, that is, it is rC .K; mn/. Recall that the sequence mrC .K; m/; m D 1; 2; : : : is an increasing one with limm!C1 mrC .K; m/ D wC .K/ and that Conjecture 7.1.9 is equivalent to the following. Conjecture 7.1.10. Let K and C be convex bodies in Ed ; d 2 and let m be P1 ; P2 ; : : : ; Pn in Ed , then Pna positive integer. If K is covered by the Pplanks n i D1 rC .Pi ; m/ rC .K; m/ or equivalently, i D1 wC .Pi / mrC .K; m/.

9.4.2 Covering Convex Bodies by Cylinders Problem 7.2.1. Prove or disprove that the sum of the base areas of finitely many cylinders covering a 3-dimensional convex body is at least half of the minimum area 2-dimensional projection of the body.

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Problem 7.2.4. Let 0 < c.d; k/ 1 denote the largest real number with the property that if K is an ellipsoid and CS k-codimensional cylinders 1 ; : : : ; CN are P N in Ed ; 1 < k < d 1 such that K N C , then i i D1 i D1 crvK .Ci / c.d; k/: Determine c.d; k/ for given d and k.

9.4.3 Kadets–Ohmann-Type Theorems Let C be a closed convex subset with non-empty interior in an arbitrary Banach space. We call C a convex body of the given Banach space. Then let r.C/ denote the supremum of the radii of the balls contained in C. One may call r.C/ the inradius of C. Problem 7.3.2. Let the ball B be covered by the convex bodies C1 ; C2 ; : : : ; Cn in an arbitrary Banach space. Prove or disprove that n X

r.Ci \ B/ r.B/:

i D1

Recall that Sd stands for the d -dimensional unit sphere in the .d C 1/dimensional Euclidean space Ed C1 ; d 2. A spherically convex body is a closed, spherically convex subset K of Sd with interior points and lying in some closed hemisphere, thus, the intersection of Sd with a .d C 1/-dimensional closed convex cone of Ed C1 different from Ed C1 . The inradius r.K/ of K is the spherical radius of the largest spherical ball contained in K. Problem 7.3.5. Let the spherically convex bodies K1 ; : : : ; Kn cover the spherical ball B of radius r.B/ < 2 in Sd ; d 2. Then prove or disprove that n X

r.Ki / r.B/:

i D1

9.4.4 On Partial Coverings of Balls by Planks Problem 7.4.1. Let Bd denote the unit ball centered at the origin o in Ed with d D 2; 4; 5; : : : . Moreover, let w1 ; w2 ; : : : ; wn be positive real numbers satisfying the inequality w1 C w2 C C wn < 2. Then prove or disprove that the union of the planks P1 ; P2 ; : : : ; Pn of width w1 ; w2 ; : : : ; wn in Ed covers the largest volume subset of Bd if and only if P1 [ P2 [ [ Pn is a plank of width w1 C w2 C C wn with o as a center of symmetry.

Glossary

aff.X / The affine hull of X Ed . ball-polyhedron (in Ed , d 3) An intersection with non-empty interior of finitely many congruent closed balls in Ed , d 3. Bd .c; r/ (resp., Bd Œc; r) The open (resp., closed) ball of radius r centered at the point c in Ed . Bd .c/ (resp., Bd Œc) The open (resp., closed) ball of unit radius centered at the point c in Ed . BŒX The intersection of the closed d -dimensional unit balls centered at the points of X Ed . bd.X / The boundary of X Ed . card./ The cardinality of a set. C.n/ The largest number of touching pairs in a packing of n unit balls in E3 . C.P/ The number of touching pairs in the unit ball packing P of E3 . Cfcc .n/ The largest number of touching pairs in a packing of n unit balls of E3 with centers in f cc . C.a; ˛/ The spherical cap of Sd 1 centered at the point a 2 Sd 1 having angular radius 0 < ˛ < . contraction of points A rearrangement of the given points such that the distances between pairs of points do not increase. contraction of spheres A rearrangement of the given spheres without changing their radii and with a contraction of their center points. convex body in Ed A compact convex set with non-empty interior in Ed . conv.X / The convex hull of X Ed . K. Bezdek, Lectures on Sphere Arrangements—the Discrete Geometric Side, Fields Institute Monographs 32, DOI 10.1007/978-1-4614-8118-8, © Springer International Publishing Switzerland 2013

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166

Glossary

convs .X / The spindle convex hull of X Ed . convex polyhedron in E3 A bounded intersection of finitely many closed halfspaces (having non-empty interior) in E3 . crvK .C/ The cross-sectional volume of the cylinder C Ed with respect to the convex body K Ed . ıd The largest density of packings of unit balls in Ed . ı.P/ The (upper) density of the unit ball packing P. deltahedron A convex polyhedron with regular triangle faces in E3 . density of a unit ball packing in Ed The proportion of Ed covered by the unit balls of the packing. diam./ The diameter of a set. disk-polygon (in E2 ) An intersection with non-empty interior of finitely many congruent closed disks in E2 . dist.; / The standard Euclidean metric on Rd . ecurv./ The edge curvature of a convex polyhedron in E3 . ec.T / The (lower) average edge curvature of the convex cells in the normal tiling T of E3 . Ed The d -dimensional Euclidean space. fat disk-polygon (resp., ball-polyhedron) A disk-polygon (resp., ballpolyhedron) that contains the centers of its generating disks (resp., balls). g-deltahedron A convex polyhedron of E3 whose each face is an edge-to-edge union of some regular triangles of side length one. globally rigid ball-polyhedron in E3 A ball-polyhedron that is uniquelly determined up to congruence by its combinatorics, i.e., by its vertex-edge-face structure and its inner dihedral angles in E3 . H Real Hilbert space. I.K/ The illumination number of the convex body K in Ed , that is, the smallest number of directions that can illuminate K in Ed . h; i The standard Euclidean inner product on Rd . int.X / The interior of X Ed . w;d

Kw;d (resp., KBL ) A convex body of constant width w in Ed having minimum BL volume (resp., surface volume).

Glossary

167

fcc The face-centered cubic lattice with shortest non-zero lattice vector of length 2 in E3 . lin.X / The linear hull of X Ed . k k The standard Euclidean norm on Rd . normal ball-polyhedron in E3 A ball-polyhedron in E3 having all the vertices of the underlying (farthest-point) Voronoi tiling in its interior. normal tiling of Ed A tiling of Ed into convex cells each containing a unit ball having a finite upper bound for the diameters of the convex cells. !d The volume of a d -dimensional unit ball in Ed . packing of convex bodies in Ed A family of non-overlapping convex bodies in Ed . plank in Ed The (closed) set of points lying between two (distinct) parallel hyperplanes in Ed . polyhedral complex in E3 A finite family of convex polyhedra such that any vertex, edge, and face of a member of the family is again a member of the family, and the intersection of any two members is empty or a vertex or an edge or a face of both members. polyhedron of E3 The union of all members of a 3-dimensional polyhedral complex in E3 possessing the additional property that its (topological) boundary in E3 is a 2-dimensional topological manifold embedded in E3 . rC .K/ The C-inradius of K, which is the factor of the largest (positively) homothetic copy of the convex body C Ed , a translate of which is contained in the convex body K Ed . rC .K; n/ The nth successive C-inradius of the convex body K in Ed . sarea./ The surface area of the boundary of a convex body in E3 . Sarea./ The spherical Lebesgue measure on S2 . d Rogers’ upper bound for the density of unit ball packings in Ed . O d K. Bezdek’s upper bound for the density of unit ball packings in Ed . Sd 1 The .d 1/-dimensional unit sphere centered at the origin o in Ed . S k .c; r/ A k-dimensional sphere of Ed centered at the point c and having radius r with 1 k d 1. Sconv.X / The spherical convex hull of X Sd 1 . simple ball-polyhedron in E3 A ball-polyhedron in E3 having exactly three edges at every vertex.

168

Glossary

spindle Œa; bs The union of circular arcs with endpoints a and b in Ed that are of radii at least one and are shorter than a semicircle. spindle convex body in Ed An intersection with non-empty interior of not necessarily finitely many congruent closed balls in Ed . s.T / The (lower) average surface area of the convex cells in the normal tiling T of E3 . simplicial ball-polyhedron in E3 A ball-polyhedron in E3 with faces bounded by three edges. standard ball-polyhedron in E3 A ball-polyhedron in E3 the vertex-edge-face structure of which is a lattice (with respect to containment). svold 1 ./ The surface volume of the boundary of a (convex) body in Ed . Svold 1 ./ The spherical Lebesgue measure on Sd 1 . d The maximum number of non-overlapping unit balls of Ed that can touch a given unit ball in Ed . m TC;d The family of all sets in Ed that can be obtained as the intersection of at most m translates of the convex body C in Ed .

unit ball packing in Ed A packing of balls of unit radii in Ed . vertex-edge-face structure of a ball-polyhedron The vertices, edges, and faces of a ball-polyhedron partially ordered by inclusion. vold ./ The Lebesgue measure on Ed . Voronoi cell (or closest-point Voronoi cell) of a unit ball packing The set of points that are not farther away from the center of the given ball than from any other ball’s center in Ed . w.C/ The minimal width of the convex body C Ed . w.C; H / The width of the convex body C Ed parallel to the hyperplane H in Ed . wC .K/ The minimal C-width of the convex body K Ed (relative to the convex body C Ed ). wC .K; H / The C-width of the convex body K Ed parallel to the hyperplane H in Ed (relative to the convex body C Ed ).

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