Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1200
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Vitali D. Milman Gideon Schechtman
Asymptotic Theory of Finite Dimensional Normed Spaces With an Appendix by M. Gromov "Isoperimetric Inequalities in Riemannian Manifolds"
Springer
Authors Vitali D. Milman Department of Mathematics Tel Aviv University Ramat Aviv, Israel Gideon Schechtman Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot, Israel
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Milman, Vitali D.: Asymptotic theory of finite dimensional normed spaces / Vitali D. Milman; Gideon Schechtman. With an appendix Isoperimetric inequalities in Riemannian manifolds / by M. Gromov. - Corr. 2. printing. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Lecture notes in mathematics ; 1200) ISBN 3-540-16769-2
Corrected Second Printing 2001 Mathematics Subject Classification (1980): 46B20, 52A20, 60FlO ISSN 0075-8434 ISBN 3-540-16769-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfi Ims or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10797471 41/3142-543210 - Printed on acid-free paper
INTRODUCTION This book deals with the geometrical structure of finite dimensional normed spaces, as the dimension grows to infinity.
This is a part of what came to be known as the Local
Theory of Banach Spaces (this name was derived from the fact that in its first stages, this theory dealt mainly with relating the structure of infinite dimensional Banach spaces to the structure of their lattice of finite dimensional subspaces). Our purpose in this book is to introduce the reader to some of the results, problems, and mainly methods developed in the Local Theory, in the last few years. This by no means is a complete survey of this wide area. Some of the main topics we do not discuss here are mentioned in the Notes and Remarks section. Several books appeared recently or are going to appear shortly, which cover much of the material not covered in this book. Among these are Pisier's [Pis6] where factorization theorems related to Grothendieck's theorem are extensively discussed, and Tomczak-Jaegermann's [T-Jl] where operator ideals and distances between finite dimensional normed spaces are studied in detail. Another related book is Pietch's [Pie]. The first major result of the Local Theory is Dvoretzky's Theorem [Dv] of 1960. Dvoretzky proved that every real normed space of finite dimension, say n, contains a (1 + c)-isomorphic copy of the k-dimensional euclidean space l~, for k = k(c,n) which increases to 00 with n (see Chapter 5 for the precise statement). Dvoretzky's original proof was very complicated and understood only by a few people. In 1970 Milman [MIl gave a different proof which exploited a certain property of the Haar measure on high dimensional homogeneous spaces, a property which is now called the concentration phenomenon: Let (X, p, J.L) be a compact metric space
(X, p) with a Borel probability measure J.L. The concentration function a(X, c), c > 0, is defined by
a(X,c) = 1- in!{J.L(A.)j J.L(A) 2:
~,
A
~X
Borel}
where
A.={XEXj
p(x,A)~c}.
It turns out that for some very natural families of spaces, a( X, c) is extremely small. For
example, it follows from Levy's isoperimetric inequality that for the Euclidean n-sphere sn, with the geodesic distance p and the normalized rotational invariant measure J.L,
a(sn,c)
~
jiexp(-c 2 n/2) .
It follows from this inequality (see Chapter 2) that any nice real function on sn must be very
close to being a constant on all but a very small set (the exceptional set being of measure
VI
of order smaller than exp( -c 2 n/2)). This last property is what is called the concentration phenomenon. It has proved to be extremely useful in the study of finite dimensional normed spaces. Going back to the concentration function, we define a family (Xn , Pn, J-Ln) of metric probability spaces to be a Levy family if a(Xn , Cn diam X n ) n~ 0 (diam X n is the diameter of X n ). Chapter 6 below contains a lot of examples of such natural families. Many of these examples have deep applications in the Local Theory. It is usually quite a difficult task to establish that a certain family is a Levy family, the methods are different from one example to the other and come from diverse areas (including methods from differential geometry, estimation of eigenvalues of the Laplacian, large deviation inequalities for martingales, isoperimetric inequalities). Levy families, the concentration phenomenon and their applications to the asymptotic theory of normed spaces are the main topics of the first part of this book. We have already mentioned one application, namely Dvoretzky's Theorem. In the same direction we deal with estimation of the dimension of euclidean subspaces of various large families of normed spaces, that is, with the evaluation of the function k(c, n) mentioned above when restricted to some (wide) families of normed spaces. (This study originated in [MIl and [F.L.M.].) Here is an example: There exists a function c(c)
>
=
>
0, such that if X n
= (m. n , II . II)
is a family of normed
0 and n, X n contains a (1 + c)-isomorphic [c(c)n"], or for any integer k and any c > 0, there is an n such that
spaces, then either for some a copy of l~ with k
0, c
>
0 and any c
>
X n contains a (1 + c)-isomorphic copy of l~. (The proof of this result uses, besides the concentration phenomenon, also the notions of type and cotype introduced below.) We also deal, in the first part, with packing high dimensionall;, 1 l~
2, spaces into
(Chapter 7) as well as packing spaces with special structure (unconditional or symmetric
bases) into general normed spaces (Chapter 10). The second part of the book revolves around the notions of type and cotype and the relation of these notions to the geometry of normed spaces. For 1 ::; p ::; 2 ::; q ::; 00 the type p constant (resp. cotype q constant) of X denoted Tp (X) (Cq (X)) is the smallest constant in the inequality
(
~ve
£.-±l
( for all k and
Xl,."
,Xk
k
II L c;x;11 2 ;=1
) 1/2
( ::;
T
k
L II x;II
) lip P
;=1
)
EX.
These notions were introduced by Hoffmann-Jorgensen [H-J] for the study of limit theorems for vector valued random variables, and were studied extensively by Maurey and Pisier ([M.P.] in particular) in connection with the geometry of normed spaces. They have proved
VII
to be a very important tool in the Local Theory. In particular, Krivine, Maurey and Pisier showed that for Px = sup{pj Tp(X) < oo} (resp. qx = inf{qj Cq(X) < oo}), l;x (resp. l;x) are (1 + e:)-isomorphic to subspaces of X for all nand e: > 0 (here X is infinite dimensionalj there is also a corresponding statement for finite dimensional spaces). We present a proof (somewhat different from previous proofs) of this theorem in Chapters 12 and 13. (Chapter 11 deals with some infinite dimensional combinatorial methods needed in the sequel.) Chapter 14 is devoted to the work of Pisier [Pisl] estimating the norm of one specific projection called the Rademacher projection. This is closely related to the relation between the type p constant of X and the cotype q constant of X· (~+ = 1). It also has applications to finding well complemented euclidean sections in normed spaces. These applications due to
t
Figiel and Tomczak-Jaegermann [F.T.J are discussed in Chapter 15. The book also contains five appendices, the first of which is written by M. Gromov and gives an introduction to the theory of isoperimetric inequalities on riemannian manifolds. It is written in a way understandable to the non-expert (in Differential Geometry). This appendix contains also results which were not published elsewhere (in particular - the Gromov-Levy isoperimetric inequality). We are indebted to M. Gromov for this excellent addition to our book.
CONTENTS Introduction Part I: The concentration of measure phenomenon in the theory of normed spaces
2.
Preliminaries The isoperimetric inequality on
3.
Finite dimensional normed spaces, preliminaries
4. 5.
Almost euclidean subspaces of a normed space Almost euclidean subspaces of l~ spaces,
1.
sn-l
and some consequences
1 5 9 12
of general n-dimensional normed spaces, and of quotient of n-dimensional spaces
19
6. 7.
Levy families Martingales
27 33
8. 9.
Embedding l;' into l~ Type and cotype of normed spaces, and some simple relations with geometrical properties
42
10.
51
Additional applications of Levy families in the theory of finite dimensional normed spaces
60
Part II: Type and cotype of normed spaces 11. 12. 13.
Ramsey's theorem with some applications to normed spaces Krivine's theorem The Maurey-Pisier theorem
69 77 85
14. 15.
The Rademacher projection Projections on random euclidean subspaces of finite dimensional normed spaces
98 106
x Appendices I. Isoperimetric inequalities in Riemannian Manifolds II.
by M. Gromov
114
Gaussian and Rademacher averages
130
III. Kahane's inequality
134
IV. Proof of the Beurling-Kato Theorem 14.4
137
V. The concentration of measure phenomenon for Gaussian variables
140
Notes and Remarks
144
Index
149
References
151
PART I: THE CONCENTRATION OF MEASURE PHENOMENON IN THE THEORY OF NORMED SPACES
1. PRELIMINARIES. In this chapter we present some preliminary material on invariant measures needed in
later chapters. Not all the details of the proofs are given. 1.1.
Let (M,p) be a compact metric space and let G be a group whose members act as
isometries on M, i.e. for 9 E G, t, s E M, p(gt, gs) = p(t, s). We begin by introducing the Haar measure. The proof we give is due apparently to W. Maak (see [Do]). THEOREM: There exists a regular measure J.t on the Borel subsets of M which is invariant under the action of members of G, i.e., J.t(A) = J.t(gA) for all A ~ M, 9 E G. Alternatively, f f(t)dJ.t(t) = f f(gt)dJ.t(t) for all 9 E G and f E C(M). (C(M) is the linear space of all real eontinuous functions on M). PROOF: and n~ =
(B(t,E:)
For each E: > 0 let
IN~I,
= {s E Mjp(t,s)
For
N~
be a minimal E:-net in M, Le., UtEN,B(t,E:) = M
the cardinality of N~, is minimal among all sets with this property. ~
E:}).
f E C( M) define J.t~(J) = n;l
L
f(t) .
tEN, Since {J.t~}~>o is a uniformly bounded set of linear functionals on C(M), it follows (see e.g. [D.S.]) that, for some sequence E:; ;~ 0,
is
a linear positive functional on C(M) with J.t(l) = 1. That is, J.t is given by a regular Borel probability measure (see [D.S.] again).
for all f E C(M) where J.t(-)
Next we want to show that the measure J.t is uniquely determined, i.e., if J.t~ is defined using a different minimal E:-net, then for the same sequence E:;, J.t~;(J) -+ J.t(J). If N~ is another minimal E:-net in M then we claim: there exists a one to one and onto map rp: N~ -+ N~ with
p(t, rp(t))
~
2E: for all t E
N~.
2
To show this we use a combinatorial result known as the "marriage Theorem" (see e.g.[Do]). Let us say that t E N£ and SEN; are acquainted if B(t,e) nB(s,e) i- 0. Then the members of any subset K of N£ are collectively acquainted with a subset L of N; of at least as many elements as K. Indeed, let L= {s E N;;B(s,c) n (UtEKB(t,e)) i- 0} then ILl ~ IKI since otherwise L U (N£ \K) is an e-net with less then jN£1 elements. The marriage Theorem states that in such a situation (where every subset of N£ is jointly acquainted with at least as many elements of N;) there is a one to one map tp: N£ -+ N; with t and tp(t) acquainted for all t E N£. This translates back into B(t,e) n B(tp(t), e) i- 0, Le., p(t,tp(t)) ~ 2e. It follows that, if JL~ is defined using N; in an analogous way to JL£, then
IJL£(J) -
JL~(I)I ~:
L
If(t) - f(tp(t))1
£ tEN.
~ w(2e)
where W(e) = sup{lf(t) - f(s)ljp(t,s) ~ e} is the modulus of continuity of f.
Thus,
limi-+oo JL~i (I) exists and is equal to JL(I).
N;
It remains to show that JL is invariant under G. Let 9 E G and let N; is a minimal e-net and
JL(J
0
g)
=
lim 1-+00
= lim
1-+00
~
L
n~i tEN.
=
f(gt)
JL~, (I) •
= JL(I)
,lim 1-+00
cra
~
L
= (gt)tEN•.
Then
f(s)
net sEN'
~i
.
o 1.2. If the action of G on M is transitive, Le. for all t, s E M there exists agE G such that gt = s, then M is called an homogeneous space of G. Fix atE M and let
Go={gEGjgt=t} then Go is a subgroup of G (called an isotropic subgroup) and M identified with the equivalence class gGo for 9 such that gt
= G/G o where
s E M is
= s. rn.n, n (rn. ,{-,.}). We
To illustrate the definition we give a simple example: Fix an inner product (".) on
Ixl = (x,x)l/2
=
On be the orthogonal group on identify On with the set of n tuples (el,'" en) of orthonormal vectors (Fix one orthonormal basis (e~, ... , e~) then any orthogonal operator A uniquely determines another such n-tuple: (Ae~, ,Ae~)). Let M = sn-l = {x E rn.njlxl = I} and tp:On -+ sn-l be defined by tp(e!> ,e n ) = el. Then clearly for any t E sn-l;tp-l(t) can be identified with On-I. So sn-l = On/On-I.
let
for x E
rn.n and
let G
1.3. THEOREM: If (M,p) is a compact metric homogeneous space of the group G then the
measure of Theorem 1.1. is unique up to a constant. PROOF: Define a semi metric on G by d(g,h) = SUPtEMP(gt,ht). Identifying elements whose distance apart is zero, we get a group H which still acts as isometries on M and also
3 on itself. (there are two ways in which H acts on itself - we choose multiplication on the right
hh' = h' . h where" ." is the multiplication in the group). One checks that H is compact (actually in all our applications G II
=H
will be given as a compact group). Let IL on M and
on H be measures invariant under the action of G. Then, for all
By transitivity of G on M and invariance of
f
E C(M),
the inner integral on the right depends on
II,
f
but not on t. Call it v(J). Then
So that if IL' is another invariant measure on M then
o 1.4. REMARKS: a. The proof shows also that any right invariant normalized measure on
a compact metric group G is equal to any normalized left invariant measure. b.
It is easily checked that the unique normalized invariant measure on G is also in-
vertable invariant i.e.
fa f(t)dIL(t)
fa f(t-1)dIL(t).
=
In what follows IL will denote the normalized Haar measure on the space in question so that it may appear twice in the same formula denoting measures on different spaces. 1.5.
We pass now to several examples of homogeneous spaces of the group On of all n x n
real orthogonal matrices. EXAMPLES: a. sn-l = {(Xl,""X n ) E IRniL:7=lx; = I} with either the euclidean or geodesic metric is easily seen to be equivalent to On/On-l as was discussed above (1.2.).
b. The Stiefel manifolds. For 1 Wn,k
=
{e
=
~
k :S n
n (e!> ... ,ek))ie; E IR , (e;, ej)
with the metric p(e, f)
= (L:7=1 d(e;, /;)2) ~,
metric. Note that Wn,n
=
On, Wn,l
=
= o;j,l:S i,j:S k}
d being either the euclidean or the geodesic
sn-l and Wn,n-l
=
SOn
=
general Wn,k may be identified with On/On-k via the map rp:On
{T E Oni det T -+
=
I}. In
Wn,k, rp(el, ... ,e n ) =
(el ... ek).
c. The Grassman manifolds Gn,k, 1 :S k :S n, consists of all k dimensional subspaces of IR n with the metric being the Hausdorff distance between the unit balls of the two subspaces
p(c,~)=
sup :l:Esn-ln~
p(x,sn-1nc).
4
The equivalence Gn,k = On/(Ok x On-k) is again easily verified. d.
If G is any group with invariant metric p and Go is a subgroup we may define a
metric d on M
= G/G o
by
d(t,s)
= inf{p(g,h)jtp(g) = t,tp(h) = s}
where tp: G ---; M is the quotient map. In this way M becomes an homogeneous space of G. Note that in all previous examples the metric given on the homogeneous space of On is equivalent, up to a universal constant (not depending on n), to the metric given here. 1.6. The uniqueness of the normalized Haar measure allows us to deduce several interesting consequences. a. The first remark is that for any A
~
sn-l and
Xo
E sn-l.
JL{T EOn; Tx o E A} = JL(A). b. Next we give two identities. Fix I the (k - I)-dimensional sphere of E. Then
~
k ~ n, for E E Gn,k we denote S(E) = sn-l
nE
for all f E c(sn-l) where JLe is the normalized Haar measure on S( E) (by our convention JL on the left is the normalized Haar measure on sn-l and on the right on Gn,k' We identify m2n with (J: n (by introducing a complex structure in one of the possible ways). For each k we denote the collection of complex k-dimensional subspaces of (J: n by
Gn, k and the unit sphere of any E E (J: Gn,k by (J: S( E) (which can be identified with S2k-l). (J: Gn,k is again an homogeneous space and we get an identity similar to the previous (J:
one
Jr
S2n-l
fdJL =
J J (J:
Gn.k
(J:
sW
f(t)dJLe(t)dJL(E).
Gn, k, than the one in the first identity which, adjusted to the dimensions here, would be G 2n ,2k' Note that here one integrates on a much smaller space,
(J:
2. THE ISOPERIMETRIC INEQUALITY ON sn-l AND SOME CONSEQUENCES 2.1.
We begin with the statement of the classical isoperimetric inequality on the sphere. A
proof is given in Appendix 1. A simple proof of a version of Corollary 2.2. (which is sufficient for the applications here) is given in Appendix V. For a set A in a metric space (M,p) and e> 0 we denote A£ = {tiP(t,A) ~ e}. In what follows we use the geodesic metric on sn-l. THEOREM: For each 0 < a < 1 and e > 0, min{J.L(A£); A ~ sn-l, J.L(A)
= a}
exists
and is attained on A o - a cap of suitable measure (i.e., A o = B(xo,r) for any Xo E sn-l and r such that J.L(B(r)) = a, where B(r) = B(xo,r) = {XiP(X,X o) ~ r}).
Using this for the case a = 1/2 (in which case A o is half a sphere) we get:
2.2. COROLLARY: if A ~ sn+l with J.L(A) 2: 1/2 then J.L(A£) 2: 1- V7r/8e-£'n/2 PROOF:
By theorem 2.1 it is enough to evaluate J.L(B(7r /2
+ e)).
Note that cosnO is
proportional to the n-volume of Se - the set of points on sn+l which are of distance 0 from
B(1/2) (this is an n-dimensional sphere of radius cos 0). It follows easily (draw a picture) that
h(e, n)
cos n OdO /17
= J.L(B(7r /2 + e)) = 1£ -7
-7
Let In = J;/2 cos n OdO and use the change of variables 0 = r /..;n and the inequality cos t ~ e- t '/2, 0 ~ t ~ 7r/2, to get
This computation (with e
= 0)
also gives
notice that integration by parts gives h Therefore,
and
..;n In
~ V7r /2. To evaluate In from below
= ((k-1)/k)h_2 which implies Vk h
2: ,jk"=2 h-2·
6 1- h(e,n) ~ ~ e-e n/2 . 2
o REMARK: It can actually be checked that 2.3.
..;n In n~ V
7r
/2.
The next corollary and theorem are crucial for the applications of the isoperimetric
inequality to Banach spaces. For a continuous function Wf(e)
= sup{lJ(x) -
mean) of
I,
I on sn+l we denote by
Wf(e) its modulus of continuity,
l(y)l;p(x,y) ~ e}. We denote by Mf the median (also called the Levy
Le., Mf is a number such that both
COROLLARY (Levy's lemma): 2 J.L(A e ) ~ 1 e-e n/2.
Let IE C(sn+l) and let A
=
{x;!(x)
= Mf}
then
V1rfi
PROOF: Note that A e
= (J
~ Mf)e
n (J
~ Mf)e
and use 2.2.
o Notice that the values of Ion A e are very close to Mf. Indeed, if e is such that Wf(e) ~ 8 then I/(x) - Mfl ~ 8 on A e . So the content of the previous corollary is that a well behaved function is "almost" a constant on "almost" all the space. This phenomenon of concentration of measure around one value of the function will appear over and over again in these notes.
2.4.
In the next theorem we trade off the set of large measure on which the function is
almost constant .with a set with linear structure - a subspace of large dimension. The symbol
[.J denotes the integer part function. THEOREM: For e,f) > 0 and an integer n let k(e,f),n) = [e 2 n/(2log 4/f))J. Let IE c(sn+l) then, for all e,f) > 0, there exists a subspace E ~ IR n + 2 with
=k
dim E
(i)
~ k(e, f),
= sn+l n E
n) and a f) - net N in SeE)
I/(x) - Mfl ~ Wf(e)
for all
such that
x EN
and x E En sn+l.
for all
(ii) follows from (i). The proof of (I) consists of the following two lemmas. As in 2.3. A
= {xjf(x) = M f }·
2.5.
LEMMA: For any N ~ sn+l with
INI < V1rfi ee
2
n/2 there exists aTE On+2 such
that TN ~ A e . Consequently, for all x E TN, I/(x) - Mfl ~ Wf(e).
PROOF: The lemma follows immediately from 1.6.a: for each x E sn+l J.L{T E On+2; Tx E A e }
= J.L(A e ) ~ 1 - V1rfi e-e
2
n/2. Therefore,
7 JL{T E On+2i Tx E A£ for every x E N} 2: 1-INlv1l"/2 e-£'n/2
>
°. o
2.6. For later use we state the next lemma in a more general framework. LEMMA:
S(X) =
For every normed space X with dim X = k there exists a 0 - net {x E Xi Ilxll = 1} with INI ~ (1 + 2/0)k ~ e k log3/8.
{Xdi::l
PROOF: Let
i
i= j.
be a maximal set in S(X) with the property that
Ilxi -
N
In
xiII 2: 0 for
Then {xi}f=l is an 0 - net. The open balls B(Xi,O/2) are pairwise disjoint and are all
contained in B(O, 1 +0/2). Comparing the volume of B(O, 1 + 0/2) with that of Uf=lB(Xi, 0/2) we get
or
o 2.7.
REMARK: An inspection of the proof shows that, taking a bit smaller k in Theorem
2.4. one can get the conclusion for most subspaces of dimension k : For k ~ [e: 2 n/(410g 4/0)] the measure of the set Ek <;;; G n + 2,k of all k dimensional subspace for which the conclusions of Theorem f!.4 hold satisfies JL(Ek) 2: 1 e-£'n/4.
J1rfi
2.8. We now indicate briefly a different way to prove Theorem 2.4. First we estimate the measure of a cap in Sk-l from below
JL(B(O))
=
1. 8 sin k- 2 tdt 0
2 h-2
>
k J:/2 sin -
-
2
2 h-2
tdt
0 sin k-
> -
2
0/2
4 Ik-2
.
Next we use the identity 1.6.b to get that there exists a k-dimensional subspace E such that
If 2
Osink- 0/2 ----,::-----'-+1 4 h-2
V
~/2 -£'n/2 > 11" / ~ e -
1
then any ball of radius 0 in E intersects A£ and then for any x E E, If(x) - MIl ~ w(e: + 0). Now use (*) to get an estimate on k. 2.9.
REMARK:
Note that if we want to prove a theorem similar to 2.4. in which the
conclusion holds for two functions simultaneously (on the same subspace) we loose' very little in the estimate on the dimension. Actually, by Remark 2.7., one can find a subspace E
8 with dim E = k 2 [e 2 n/(4Iog 4/0)] for which the conclusion holds for exp ([e 2 n/(4Iog 4/0)]) functions simultaneously. 2.10. We conclude this chapter with a few exercises.
a. Let A ~ sn-l. Then either A~ or (AC)~ contains sn-l nE for some k(e/2,e/2,n)dimensional subspace E. (k(e,O,n) was defined in theorem 2.4.). b. Fix k ~ k(e/2,e/2,n), let A ~ sn-l be such that for all k-dimensional subspaces E, An E
=1=
0. Then there exists a k-dimensional subspace E, with sn-l n E
~ A2~'
HINT: Consider the function p = p(x, A) and prove that the median M p is at most e. c. For every A ~ sn-l we define
and IA,k = {E E Gn,k; S(E) ~ A} . Fix e > 0 and k ~ [e 2 n/10 loge-I].
Prove: if J.t(BA,k) > 2e-~2n/4 then J.t(IA ••.• ) 2
1 - 2e-~2n/4.
HINT: First use Remark 2.7 to show J.t(A2~) 21/2.
3. FINITE DIMENSIONAL NORMED SPACES, PRELIMINARIES Let X, Y be two n-dimensional normed spaces. The Banach-Mazur distance between
3.1.
them is defined as
d(X,Y) Obviously d(X, Y)
~ 1
= inf{IITII·IIT- l llj
T:X
-+
Y isomorphism}.
and d(X, Y) = 1 if and only if X and Yare isometric. If d(X, Y)
~ A
we say that X and Yare A- isomorphic. The notion of the distance also has a geometrical interpretation. If d(X, Y) is small then in some sense the two unit balls B(X)
=
{x E Xj II xii ~ I} and B(Y) = {y E Yj lIyll ~ I} are close one to the other. More precisely there is a linear transformation ep such that B(X)
~
ep(B(Y))
~
d(X, Y)B(X).
The Banach-Mazur distance satisfies a multiplicative triangle inequality (d(X, Z) ~ d(X,Y)· d(Y,Z)). Also d(X",Y") = d(X,Y) for all X and Y where" denotes the dual space. In the next few chapters we will consider the space mn with two norms on it. One is a general norm 11·11, the other will always be an euclidean norm Ixl = (x, x) 1/2 induced by some inner product (., .). We denote D = {x E mnj Ixl ~ I} and for E ~ m n , S(E) = {x E Ej Ixl = I}. Let a, b be such that 3.1.1. We may define the norm, dual to
II· II
relative to (-,.), by
We get easily that
n
3.1.2. b-llxl ~ Ilxll" ~ alxl for all x E m . Indeed, Ixl2 = (x, x) ~ IIxllllxll" ~ blxlllxll", which gives the left side inequality. On the other hand for all x,y E mn\{O}
l(x,y)1 < ~ <
Ilyll -
so that
Ilxll"
= sup
{I(II~II)I
j
y
Ilyll -
alxl
=f o} ~ alxl-
3.2. The following two theorems deal with the ellipsoid of maximal volume inscribed in the unit ball of a normed space. We recall that a centrally symmetric ellipsoid in IR n is the body
10 obtained as the image of the unit ball under linear transformation. Equivalently an ellipsoid is the unit ball of any euclidean (Le. given by an inner product) norm in m. n • The uniqueness of the ellipsoid of maximal volume holds but not trivially. Since we are not going to use it we shall not prove it. 3.3. THEOREM (F. John). Let X = (m n , II ·11) be an n-dimensional normed space and let D be the ellipsoid of maximal volume inscribed in B(X). Let 1·1 be the euclidean norm
induced by D (i.e. D
= {Xi Ixi
::; I}). Then
(l/y'n)lxl::; Ilxll ::; Ixl· Consequently, the norm
d(X,l~)
::;
I(xl> ... xn)1 =
yn
(l~
is the n-dimensional canonical Hilbert space, i.e., (L~=l xW/ 2).
mn
with
We only sketch the proof briefly. Since D ~ B(X), Ilxll ::; Ixl. We want to show that B(X) ~ y'n D. Applying a linear transformation, we may assume that D = {(XI""'X n ) E m.nj L~=lx; ::; I}. If
B(X) ~ ..;n D then there exists apE B(X) with Ipl > ..;n. Since B(X) is convex, also K = conv D U (±p) ~ B(X). We want to show that K contains an ellipsoid of volume larger than the volume of D, in contradiction to the assumption. Without loss of generality we may assume that p = (d, 0, ... ,0), d > ..;n. The ellipsoid n
{x E
m. n jxUa 2 + LxUb2::; I} i=2
has volume a· bn -
Vol (D) and as long as a 2/d 2 + (1-1/d 2)b2 ::; 1 it is contained in K (check). The pair a = d/..;n, b = l/n / 1/ d 2 satisfies this last requirement and a' bn - l > 1. l
.
VI -
VI -
o 3.4. THEOREM (Dvoretzky-Rogers): Let X = (R n , 11·11) be an n-dimensional normed space
and let D be the ellipsoid of maximal volume inscribed in B(X). Then there exists a basis (Xi)~=l orthononnal with respect to D such that 1 ~
IIxili ~ 2- n /(n- i+l), i = 1, ... , n -
PROOF: We choose an orthonormal system inductively in the following way:
x;).L n D with maximal possible
Ixl::; 1, IIxii
is any
IlxIiI = 1). Given XI, ... ,Xi we choose Xi+! in II ·11 -norm. Then for any X E span(xi,"" x n ) with
vector in D with maximalll·ll-norm (clearly (Xl' ••• '
Xl
1.
::; Ilxill·
Consider the ellipsoid
~i-l 2 ~n 2 } wI aj +~<1 . a2
b2
-
11
Ei'=i ajXj E bD and thus II Ei'=i ajxjll ::::: bllxiII- Choosing a = 1/2, b = 1/(2I1xill), we get that E ~ B(X). On the other hand Vol (E) = 2-(i-l)(2I1xiID-(n-i) Vol (D), so necessarily 2- n+ 1 1Ixill-(n-i) ::::: 1 or II xill ~ 2-(n-l)/(n-i). o
4. ALMOST EUCLIDEAN SUBSPACES OF A NORMED SPACE. As in Chapter 3, let X = (m n+2, 11·11) be an (n+2)-dimensional normed space and let be an euclidean norm on
m
n
1·1
2
+ satisfying 3.1.1.
We are going to apply the results 2.3. and 2.4. to the function r(x) = Ilxll defined on mn+2 ; Ixl = I}. Note that the modulus of continuity of r satisfies wr(e) :S b· e.
sn+l = {x E 4.1.
Inspecting the statement of Theorem 2.4., it seems that if we want r(x) to be close to
its median M r we need to choose both e and 0 of order Mr/b. The next lemma shows that, due to the fact that r(x) is the restriction of a convex and homogeneous function, we can do a little better: 0 can be chosen independently of Mr/b. This will be important in some of the applications especially in the proof of Dvoretzky's Theorem.
Assume Ir(x) - Mrl :S be for all x in a O-net N subspace E of mn +2 then, LEMMA:
1 - 20 --JI
1-"
Mr
be -
--JI
1-"
:S
1 r(x) :S --JlMr 1-"
Ilxll =
of E n sn+l for some
be
+ --Jl. 1-"
for all x E En sn+l. PROOF: Let x E En sn+l, by successive approximation one can find {Y;}~1 in Nand {0;}~1 in
m with 10il :S Oi-l
such that x
= Yl
00
II x ll
:S
IIYil1 +
+ L~2 OiYi. It follows that 1
00
L OillYil1 :S L Oi(M
r
+ be) = 1 _ 0 (Mr + be).
i=O
i=2
To get the lower estimate, let yEN be such that
Ilx - yll
Ix - yl
:S o. Then, by the first part,
:S l~e (M r + be) and
Ilxll 2 Ilyll-llx - yll 2
Mr
-
be -
o -JI
1-"
(M r
+ be)
1 - 20 = --"M r
1-"
1
-
-"be. I-v
o 4.2. We now apply Theorem 2.4. and the Lemma above to get an estimate on the dimension of a subspace of
mn + 2
on which the norm
II . II, satisfying 3.1.1.,
is almost a multiple of the
euclidean norm.
II
THEOREM: For any 0> 0 there exists a c(o) > 0 such that for any n and any norm . lion mn+2 satisfying 9.1.1. there exists a subspace E of mn+2 such that k = dim E 2
13
c(o) . n . (Mr /b)2 and (1 - 0) . Mr' d(E,l~) ~ (1 + 0)/(1- 0).
Ixl
~
Ilxll
~ (1
+ 0)
. Mr .
Ixl
for x E E. In particular
PROOF: Let 0 > 0 be such that
1
0
1 - 20
0
1-0
2
1-0
2'
- - < 1 + - and - - > 1 - let
0:'
> 0 be such that 1 + e;' 1 8 nd 1 - 29 - e;' 1_ 0 1-9 < + a 1-9 > ,
*
and let
c(o)
0:
=
£I.~r. By Theorem 2.4., there exists an
and a O-net N in En
sn+1
E
~
n
R + 2 with dim E;:::
[~ lo~2:18]
=
such that
for all x EN. It follows from Lemma 4.1. that, for all x E E,
o REMARKS: a.
It is easy to see from the proof that the constant c(o) satisfies c(o) ;:::
c0 2/10g% for some absolute constant c. Using Remark 2.7. one may show that the collection of all subspaces E
b.
having the property d(E, l~) ~ (1 4.3.
+ 0)/(1 -
~
G n + 2 ,k
0) has measure;::: 1 - exp( -c(o) . n . M; /b 2 ).
We turn now to the problem of finding large euclidean sections in a finite dimensional
normed space. Theorem 4.2. reduces this problem to the problem of finding euclidean norms for which Mr/b is as large as possible. For a finite dimensional normed space X and for we denote by k(X,o:) the dimension of the largest Hilbert space which is (1
0:
> 0,
+ o:)-isormorphic
to a subspace of X. THEOREM: Let X = (m n , II·
III
and assume d(X,l~) ~ d n , then
k(X, 0:) where c(o:) ;::: c· 0: 2/1 0g
Ilx'l
c(o:)· n/d~
c an absolute constant.
There exists an euclidean norm on IR n such that a . b
PROOF:
a-llxl
i,
;:::
blxl
for x E IR n . Obviously M r ;::: d;;l. Now apply Theorem 4.2. ~
~
min{llxll; Ixl = I}
;:::
a-l
~
d n and
so that Mr/b;:::
o
14 If for a family of spaces {Xn}::"=l we have sUPn d(Xn , (2) = M < 00 then we get from the Theorem above that k(Xn,e)/n is bounded from below by a constant depending on e and M alone. For this reason it is enough (for the purpose of getting asymptotic estimates) to deal only with k(Xn , 1) (say). We shall denote k(X, 1) by k(X). Before continuing with the investigation of the quantity Mr/b for some special families of spaces, we would like to gather some facts about the relation between k(X) and k(X·). 4.4. X
Ilxll',
Applying Theorem 4.2. to the dual norm r'(x) =
= (m. n +2 , II . II)
we get from 3.1.2. that if
and xE
then there exists a subspace F ~
m. n +2 with
(1 - e5) . M .. ·Ix\
~
Ilxll'
k'
= dim
m. n + 2
F ~ c(e5) . n· (M.. /a)2 and
(1 + e5) . M ..
~
'Ixl
for x E F.
Moreover, if k ~ c(e5) . n' min{(Mr /b)2, (M.. /a)2} then Remark b in 4.2. implies that one can find a subspace E ~ m. n with dim E = k such that simultaneously
and
(1- e5) . M ..
·Ixl
~
tlxll'
~ (1
+ e5) . M .. 'Ixl
for all x E E.
We summarize these two remarks (specifying a e5) in the following theorem. THEOREM: There exists an absolute constant c such that for every normed space X =
(m n , II . II)
and euclidean norm
I· I on mn
satisfying
we have
(i)
~ c. n2 . (Mr. M .. ) 2
k(X) . k(X')
a·b
and
(ii) There exists a subspace E
~
mn
with dim E ~ c· n' min{(Mr /b)2, (M.. /a)2} such
that 2
3" . M r 'Ixl
~
II xii
~
4
2
3" . M r 'Ixl and 3"' Mr. 'Ixl
4.5. Note that always Mr' M .. an x E sn-l with
Ilxll
~ M r and
~
~
Ilxll
•
~
4
3" . M .. ·Ixl, x
E
E.
1. Indeed, by the definition of M r and M .. , there exists
Ilxll'
~ M ... Then,
Mr' M .. ~
Ilxll' 'lIxll
~ (x,x) = 1.
15 Next, one may choose 4.4.
1·1 in such a manner that a·b =
d(X,q), therefore we get from Theorem
COROLLARY:
In particular,
k(X) . k(X') 2: c· n . The last statement follows from Theorem 3.2. 4.6. The next proposition evaluates the norm of a projection onto the euclidean section one
obtained in Theorem 4.4. (ii). PROPOSITION: Let E ~
mn
be a subspace sa'tisfying (ii) of Theorem 4.4. (all is needed
is Ilxll :::; ~ . M r 'Ixl, Ilxll* :::; ~ . M" 'Ixl x E E). Let P 1·1), projection on E. Then 16 IIPII :::; 9 . M r . M r *
(IIPII
be the orthogonal (with respect to
is the norm of the projection as an operator from X to X).
PROOF: For any x E IR n ,
IPxl
2
= (Px, Px) = (Px, x)
:::;
4
IIPxll" . Ilxll :::; 3' . M" . IPxl . IIxll.
Thus,
IPxl :::;
4
3' . M" . Ilxll
and 4
IIPxl1 :::; 3' . M r
·IPxl :::;
16
9' Mr' M" ·lIxll· o
4.7. Putting together Theorem 4.4. and Proposition 4.6. we get the inequality 4.7.1.
k(X) . k(X*) 2: c· (n
·IIPII/(a· b))2
where P is a projection onto a subspace satisfying (ii) of Theorem 4.4. Moreover, choosing the ellipsoid that gives the distance of X to l2, we get
4.7.2.
k(X)· k(X") 2: c· (n '1IPII/d(X,l2))2 .
= (IR n , 11·11) be a normed space and let 1·1 be an euclidean norm on IRn satisfying 3.1.1., a-llxl :::; Ilxll :::; blxl. We related above the search for euclidean subspaces of X with the median M r of r(x) = IIxll on the sphere sn-l = {x; Ixl = I}. The next theorem 4.8. Let as before, X
relates this search of subspaces of X, closer to being euclidean then X itself, to the median of r*(x) = IIxl'". In the next chapter we shall use this information to obtain good euclidean subspaces and quotient spaces of X.
16
THEOREM: Let X satisfy 3.1.1. Then for any 0 < ¢J < 1 there exists a subspace E
~
X
with k
= dim
E 2: (1 - ¢J)n
and
for all x E E where c > 0 is an absolute constant.
For the proof we need two lemmas the first of which is a different version of 2.2.
i
4.9. LEMMA: a. Let A ~ sn-1 with JL(A) 2:
and let 0 <
g
<
~. Then for some absolute
constant c > 0,
b. Let 0 < 6 <
(recall that B(6)
~
then for some absolute constant
C1,
= B(xo,r),xo E sn-1).
PROOF: a. With the notation of 2.2,
JL(A i - e ) 2: JL(B(7l" - g))
=1> -
_1_
21n -
2
=1-
t
JL(B(g))
sin n -
10
.;n=-2 .
1- - - - sm
n-2
2
2
ede
g.
b. Put t = (n-~J37' then,
21n -
2
sin n -
2
ede
5
1
> -- 21 n
1 5
1
JL(B(6)) = - -
2
r 1
sin n -
2
ede
5 (1-t)
In-2
2: - 2 - 6t sin n - 2 (6(1 - t))
> :. sin n -
-n
1
6
o 4.10. LEMMA: Let X = (m
such that for all 0 < ¢J <. ~
n
,
Ilxll).
Then there exists a set A ~ sn-1 with JL(A) 2:
4 and
17 PROOF: Take A = {y E S,,-l i IlyliO ~ Mro}. Then for x E At-4> we can find ayE A with
dist(x, y) =
i - ¢,
that is, 11"
•
(x,y) = cos(2" - ¢) = Sln¢. Then Il
xll> (x,y) > sin¢ - IlyliO - M ro
o 4.11.
I·
PROOF of 4.8. : As usual we may assume that
I is the standard euclidean norm.
It follows from 9.5.1. below that, for some absolute constant c > 0, ,;nMro 2: cr. Also, in all the relevant applications we actually have
.JTi!vfr 2: o
r. We therefore assume in this proof
that ,;nMro 2: r and we thus trivially have the conclusion for ¢ ~ assume ¢ 2:
-jn
(and k=n) and we may
-jn.
Let A ~ S"-1 be as in Lemma 4.10. and fix an integer k = A(n - 2),
! < A < 1.
Then,
averaging over all k + 1 dimensional subspaces of IR" and using Lemma 4.9. we find a (k + 1}dimensional subspaces Ek+l ~ IR" with
4.11.1.
NOTE: Changing c to another absolute constant the same conclusion holds for a
subset of G",k+l of measure larger than
!.
This will be used in the next Chapter.
If 8 is such that 4.11.2.
Cl
_ k sink 8 + 1 -
+1
then, by Lemma 4.9., At-£ in Sk.
n Sk
c.JTi
sin"-2 c > 1
intersect any 8 - cap in Sk, that is, A t -£
n Sk
is a 8 - net
Take 8 = c - ¢ then, if 4.11.2. is satisfied, (A;--£)£-4> = At-4> covers Sk and by Lemma 4.10. we get that Ek+l satisfies the conclusion of the theorem. It remains to evaluate the largest k for which 4.11.2. is satisfied. Rearranging 4.11.2. and taking (n - 2) - th root we get that 4.11.2. is satisfied if
with
j.L"
< (-~)
*
for some absolute c.
This is equivalent to
18 or (sin f:)!.? -'------'--:-- ~ JLn (1 - tantPcotf:) cos tP or
1->'
->.-log(sin f:) - log cos tP ~ logJLn
Fix f:( =
i,
+ log(l -
tantPcotf:)
say) and notice that
and
3 n IlogJLnl ~ -2 log-« n c
llyn ~ tP·
Consequently, we get, for small tP ~ ~, that 4.11.2. is satisfied as long as 1~ A
~ etP (c
absolute)
or, as long as
>. <
1-
etP (c absolute)
This proves the theorem (use ctP instead of tP).
o
5. ALMOST EUCLIDEAN SUBSPACES OF l; SPACES, OF GENERAL n-DIMENSIONAL NORMED SPACES, AND OF QUOTIENT OF n-DIMENSIONAL SPACES 5.1. In order to use Theorem 4.2. in special spaces one has to evalute Mr/b from below. It turns out that it is much easier to evaluate the average
rather than the median Mr. The next lemma shows that typically the two quantities are close each to the other. LEMMA: There exists a constant C such that if 9.1.1. holds with b ~
If a . b ~
.,;n then
.,;n than
PROOF: By 2.3,
Thus, if b ~
.,;n,
and
To prove the last claim note that we may assume, multiplying the norm by a constant, that a ~ 1, b ~ Also M r ~ a-I ~ 1 so that by the first part
.,;n.
and
The left hand side inequality A r
~
M r /2 holds with no restriction on a or b.
o REMARKS: 1. The proof also shows that, if b ~
.,;n,
20
for all 1 :S p
<
00
with C p depending on p only.
2. If a· b = o( yin) then the proof shows that M;l . A r = 1 + 0(1). 5.2.
The family of
normed spaces.
e;
spaces, 1 :S p :S
e; is the space rn.
n
00,
plays an important role in the local theory of
equipped with the norm n
lI(xl, ... ,x n )llp=(2)xiI P)1/p
p
i=l
As is well known
(e;t = e; where l/p + l/q =
1 (1/00 = 0). We are going to apply now
Theorem 4.2. and Corollary 4.5. to the special case of 5.3.
e; spaces.
Euclidean sections of e'l
Note that by Holder's inequality for all x E
rn. n .
So that using 11·112 as the euclidean norm in 4.2. we get that b = ..;n. To evaluate M r we use Lemma 5.1. and the evaluation of In from 2.2.
Mr(e~):::: =
1 1 C- . Ar = C-
c- 1 . n·
1sn-l
::::C-1·n·
Ln-l Ilxlll dJ.L(x) = C- 1 t.=1 Ln-l IXildJ.L(x) =
IXddJ.L(x) =
c- 1 . n .
7r/2
J
cos lI(sin lIt-2dll / I n - 2
-7r/2
~1·vh(n-2)/7r::::d.,;n n-
for some absolute constant d. Therefore, using Theorem 4.2., we get
for some absolute constant 6. 5.4.
Euclidean sections of
e;, 1 < p <
00.
We shall need two facts. FACT 1: for 1
00
d(e;,e 2):s nll/p-l/21
PROOF: By Holder's inequality we get, for 1 :S p :S 2, n
n
n
i=l
i=l
i=l
(L an 1/ 2 :s (L la;IP)l/p :S nl/P-l/2(L:>nl/2 For q > 2 one gets the result by duality or by a similar inequality.
o
21 FACT 2: For q
> 2 there exists a constant C q such that k(e;)
~ C q . n 2/ q for all n.
We shall prove Fact 2 in 5.6. below.
<
Now, corollary 4.5. implies that, for 1
C q . n 2/ q . keen) p
P
<
2 and q
> 2 such that
n 2 > keen) q . keen) p -> c· n /d(e q' en)2 2 ->
-
l/p
+ l/q =
1,
c. n 1+ 2/ q •
Since clearly k(e;) < n, we get that we actually have equivalence in all the inequalities above. We conclude with three corollaries.
a.
For q > 2, k(e;)/n 2 / q is bounded from above and below by constants depending only on q.
b.
n 2': k(e;) 2': c p • n, p
c.
nll/p-l/21
< 2, cp depends only on
p
2': d(e;,e 2) 2': cp ' nll/p-l/21, 1 < P <
00,
c p depends only on p.
We remark in passing that the constant in b can actually be choosen to be independent of p (using a proof similar to 5.3.). The constant in c is known to be 1, i.e., d(
Using Proposition 4.6., one can show that
~ n 2 / q which is "nicely" isomorphic to
e2
2 /.
e2,e;) = n 1/ 2 - 1/ q •
e;, q > 2, contains a subspace of dimension'
and "nicely" complemented.
5.5. For the proof of fact 2 we need the notion of the Rademacher functions which will play an important role also later in these notes. Define, for k = 1,2, ... and 0
~
t
~
1
For example,
rICt)
={
1 -1
t < 1/2 { 1 t < 1/4, r2(t) = t > 1/2 ' -1 1/4 < t < 1/2,
1/2 3/4
< t < 3/4
This is a sequence of ±1 valued functions (disregarding a set of measure zero) which is (statistically) independent. Any other sequence with these two properties will do for our purposes. For example, it will sometimes be more convenient for us to use a sequence
(ri(t))~l
defined on the probability space {-1, l}lN endowed with the product measure obtained by assigning the measure 1/2 to each of -1 and 1, where ri(t) is defined, for t = (tl,t2," .), by ri(t) = ti
Averages of the form
i = 1,2, ....
22 with Xi in some normed space and 1 :S p < 00 will play an essential role in these notes. Note that, in view of the second definition of (ri)~l> these averages can be represented also as
As an example note that if
Xi
E H, i = 1, ... , n, H a Hilbert space, then the parallelogram
equality implies that
The following inequality regarding linear combinations of the Rademacher functions will be used in the proof of Fact 2 momentarily. It will be used over and over again throughout these notes. We postpone its proof until Chapter 7. KHINCHINE'S INEQUALITY: For 1:S p < 00 there exist constants 0< A p, B p < such that 1 laiI 2)1/2 :S I airilP d(tW/p :S laiI 2)1/2
(i
Ap(~
~
00
Bp(~
for every n and every choice of a1,'" ,an' The value of the best constants A p , B p is known ([HaaJ). We remark only (and this will follow from the proof in Chapter 7) that B p ~ v'P while A p remain bounded away from zero for all p.
5.6. Proof of fact fJ: Let U;
be vectors in
l;
= (U;,l"'"
u;,n), j
= 1, ... , k = k(l;)
such that
k
k
k
;=1
;=1
;=1
(L: la;1 2)1/2 :S II L a;u;lIq :S 2(L la;1 2)1/2 for any choice of a1,'" ,ak. Then, for any t E [0,1],
q k /2 =
k
k
(L rJ(t))q/2 :S II L ;=1
n
r;(t)u;lI~ =
k
L IL r;(t)u;,il q . i=l ;=1
;=1
Integrating over [0,1] we get, by Khinchine's inequality,
q k / 2 :S
L 1 L ri(t)u;,il q dt :S B: L(L U;,i)q/2. n
1
n
k
k
1
;=1 0
;=1
;=1 ;=1
Now, for each fixed i
k n k 2 . < ('" I'" u· ·u· .Iq) l/q = 'L..J " u ',I L..., L.J 1,1 J,~ ;=1 £=1 ;=1
k
k
;=1
;=1
I '" L..., u·1,1·UJ·11 q < - 2('" L..., u~ J,I.) 1/2.
23
So that
k
.)1/2 ( "L-J U2 ],1.
< 2 -
;=1 and from (*)
That is
o 5.7. Euclidean sections of
l~.
The proof of Fact 2 in 5.6. shows also that
k(l~)::;
C ·log n for some absolute constant
C. Indeed, since
we get that
Consequently, since B p
R:
..;p,
k(ln00""" ) - k(lnlog
< (2. B log n )2. n 2/ log n < _ C ·log n .
n) _
Using Theorem 4.2. we are going to show that as the euclidean norm in 4.2.
k(l~)
we have that b = 1.
R:
log n. Using the l2
norm
So that it is enough to prove that
M, ~ c· (log n/n)1/2. By lemma 5.1. it is enough to get a similar estimate for A,. For later
purpose we prove a more general estimate. LEMMA: For 1 ::; m ::; n,
m)
log max IXildJ.L(x) ~ c· ( - l:5.:5m n for some absolute constant c
>
1/2
In particular, for r(x) = max1:5i:5n
O.
IXil ::;
1, A, ~ c'
(log n/n)l/2.
PROOF: Let v be the measure on IR n with density exp( -'If ·2:7=1 tn. This measure is a probability measure and is clearly invariant under isometries of l2' Le., under On. If !:sn-1
-+
IR is any integrable function, define j(t)
JIRR j(t)dv(t)
= Ilt112' !(ntiT;-)
in IRn\{O}. Then
is invariant under the action of On and the uniqueness of the Haar measure
J.L on sn-1 implies the existence of a constant An such that
JIRR j(t)dv(t) =
5.7.1. Using polar coordinates and putting
! ==
An
Jsn-l !(t)dJ.L(t).
1 in 5.7.1. one sees that
24 for some absolute constant C.
fm
n
This reduces the problem to the problem of estimating
maxl~;~m It;ldv(t) from below.
Now, for any a> 0,
Choosing
0:
= C' Vlog m for some absolute c we get that the last quantity is :s 1/2 so that
and
r lm
max It;ldv(t) ~ 1/2. C' Vlog m. l~.~m Combining this with 5.7.1. and the estimate for An we get
r
1sn-l
n
max IX;ldJL(x) =
l~.~m
A~l
r
lm
n
m,ax It;ldv(t)
l~.~m
~ c(log m/n)1/2 o
5.8. We are now in a position to prove Dvoretzky's Theorem.
There exists an absolute constant c such that k(X) n-dimensional normed space X. THEOREM:
~
c . log n for any
PROOF: Let D be the ellipsoid of maximal volume contained in B(X). Let the norm associated with it. Then by F. John's Theorem 3.2. l 2
n- /
In particular, b = 1 and a' b:S
Ixl:s Ilxll :s Ixi
1·1 be
forall x E X.
..;n, so that we may use Lemma 5.1. to evaluate
Mr.
By the Dvoretzky-Rogers Lemma 3.3. there is an orthonormal basis Xl,"" x n
Ilx;11
~
1/4, i
= 1, ... [n/2].
hn-l I ~ a;x;lIdJL(a) hn-l II ~ r;(t)a;x;lIdJL(a).
Ar =
=
Note that, by the triangle inequality, for all {y;} n
1
~
X n
10 II t; r;(t)y;lIdt = Avee,=±lll t;c;y;11 ~ l~fm Ily;ll· So, integrating (*) over [0,1] and using Lemma 5.7., we get,
r 1 II t r;(t)a;x;lldtdJL(a) ~ 1sn-l ~ r max lIa;x;lldJL(a) ~ ~ r max la;ldJL(a) ~ c· (log n/n)1/2 lsn-l l~.~n 4 lsn-l 1~.~[n/2J 1
Ar
=
0
;=1
with
Now, for each fixed t E [0,1],
(*)
25 for some absolute constant c. By Lemma 5.1., we get now that M r 2 c(log n/n)l/2 for some absolute c. Finally, using Theorem 4.2. we get the desired result.
o 5.9.
We conclude this chapter with a theorem stating that every n-dimensional normed
space has a subspace of a quotient space (= a quotient space of a subspace) of dimension proportional to n which is close to being euclidean. THEOREM: Let X 'be an n-dimensional normed spaces and let A < 1. There exists a
quotient space Y of a subspace of X with dim Y
2 An and
where c is an absolute constant. PROOF: We shall need one fact which will be proved only in Chapter 15. Namely, there exists an euclidean norm r(x) =
c (and
a-1lxl :S Ilxll :S blxl
Ixl on X
with a/Mr-
for which MrMr-
:S c 10g(d(X, £2)) for some absolute
:S yn).
By Theorem 4.8. there exists, for any 0 < 4> < 1, a subspace E
~
X with dim E = k 2
(1 - c4»n (c absolute) and
xE E. Moreover, (Note 4.11.1.) this holds for a subset of Gn,k (the Grassmann manifold) of measure larger than or equal to~. Using Lemma 5.1. and averaging over Gn,k we get that for a subset of Gn,k of measure larger than or equal to~, ME,r (the median of r restricted to
BE
= {x E Ej Ixl =
I}) is smaller or equal than c M r for an absolute c. (Actually, for any E
with dim E = k, ME,r
:S
c~ Mr. This follows from 9.5.1. below. We prefer not be use it
here). Consequently, we can find an E ~ X with dim E = k 2 (1 - c4»n on which
4>
M - Ixl :S r
Ilxll :S b[xl
xEE
and
ME,.:S c Mr. Since
b-1lxl :S Ilxll' :S ~-Ixi
xE E
we can find by the same argument a subspace F of E' with
dim F 2 (1 - c4»2 n and
(c absolute)
26
In particular, d(F, ldim F) 2
r- < cA.- 21 < MrM 0g ¢2 'I'
d(X , en) . 2
Note that F* is a subspace of a quotient of X. Since log d(X,lz) Theorem up to a log n factor. To get rid of that log n, define for t f(t)
= inf{d(F,l~n)j
~ log n
=
~,k
this proves the
= 1, ... , n,
F a subspace of a quotient of X with dim F
= tn}
.
Then the proof above shows !CAt)
< -
c
1
(1 -
vxA)2 log f(t)
(for any 0 < A < 1 such that At = ~ for some k = 1, ... ,n). We may extend f to all of (*,1] in such a way that this inequality still holds. In particular, 5.9.1.
We also have 1
~ f(s) ~
n. The result follows by iterating 5.9.1. We indicate briefly how to
perform the iterations. Put g(s) = f(s)
(2 (l_cS~)210g (l_cS~)2)-1
Then, for s close enough to 1 (independently of n), 5.9.2. as long as g(s) 2: e (check).
Iterating 5.9.2. we get
where log(f)x
= log log(t-1)x and log(l)x = log
x.
Given 0 < a < 1, let t be the first such that log(t)n ~ e and take s such that S2' = a. Then, g(a) ~ log(t)g(s) ~ e
and we get the desired result.
o
6. LEVY FAMILIES
6.1. Let (X,P,JL) be a metric space (X,p) equipped with a Borel probability measure JL. For a subset A
~
X and e: > 0 we define, as in Chapter 2,
A e = {x E Xi p(x,A) ~ e:}. Let diam X be the diameter of X and assume diam X 2: 1. The concentration function a(X, e:) is defined for any e: > 0 by
a(X,e:) = 1- inf{JL(Ae)i A DEFINITION: A family (Xn,Pn,JLn), n a Levy family if for every e: > 0,
~X
Borel with JL(A) 2:
~}
= 1,2, ... of metric probability spaces is called
The family is called a normal Levy family with constants
Cll C2,
if
Note that any normal Levy family is a Levy family (since we assume diamXn 2: 1). The omission of the factor diamXn in the definition of a normal Levy family comes because that way most of the examples below become Levy families with their natural metric and natural enumeration. As we have already seen in specific examples in Chapter 2, in a Levy family we necessarily have the phenomenon of concentration of measure around one value of a function. If f : X -+ m is a function with modulus of continuity wf(e:) and with median Mf then we get
6.1.1.
JL(lf - Mfl ~ wf(e:)) 2: 1- 2a(X,e:).
Indeed, put A = {x E Xi f(x) ~ Mf}, B = {x E Xi f(x) 2: Mf} then JL(A), JL(B) 2: 1/2 and
JL(IJ - Mfl ~ wf(e:)) = JL[(J ~ Mf
+ wf(e:)) n (J 2: Mf
- wf(e:))]
2: JL(A e n Be) 2: 1- 2a(X,e:) . That is, f is concentrated close to Mf on most (in the sense of measure) of X, if a(X,e:) is small enough. In 2.2. we have shown that {sn+l }~=l' with the natural metric and probability measure, is a normal Levy family with constants
Cl
=
J1r78 and C2
= 1/2. We applied this fact in
28 Chapters 2 and 4. In this Chapter we shall give a lot of other examples of Levy families. We shall see that a number of natural families of metric probability spaces are Levy families. Most of these examples found applications in the Local Theory of normed spaces. A few of them play a central role in this theory and for them we present a full detailed proof here or in later Chapters; the rest are only briefly sketched. 6.2. Let E2: = {-I, 1}n with the (product) measure
and with the normalized Hamming metric l I n
dn(x,y)
= -1{i;Xi t- y;}1 ="2 L n
n
IXi - Yil,
x,y E
E;.
i=1
THEOREM: (E2:, d n , P n ) is a normal Levy family with constants
Cl
= 1/2 and
The theorem follows from an isoperimetric inequality (i.e. identifying the sets A
C2
~
= 2.
E2: for
which the infimum inf{P(A~); P(A) 2 1/2} is attained) of [HarJ. We shall present a different proof (giving somewhat worse constants) in Chapter 7 below. The same proof will prove also the next result of [Maj. 6.3. Let TIn be the group of all permutations of the set {I, ... ,n}. Let P n be the normalized counting measure;
Pn(A) = IAI/n!,
A
~
TIn
and let d n be the normalized Hamming metric;
THEOREM: (TIn, d n , P n ) is a normal Levy family with constants
Cl
= 2,
C2
= 1/64.
6.4. The following few examples of Levy families are consequences of a general isoperimetric inequality for connected riemannian manifolds due to Gromov [Gr1J. Appendix I, written by Gromov, contains the proof together with the necessary definitions. Let J.lx be the normalized riemannian volume element on a connected riemannian manifold without boundary X and let R(X) be the Ricci curvature of X. THEOREM: Let A
~
X be measureable and let
where B is a ball on the sphere
and J.lx(A)
= J.l(B),
T •
sn
E:
> 0 then
with n = dimX, and
T
such that
J.l being the normalized Haar measure on T·
sn.
29 The value of R(X), known in some examples ([C.E.]), together with the computation for the measure of a cap in 2.2. leads to the following examples. THEOREM: The family SOn = {T EOn; detT = 1}, n = 1,2 ..., with the metric discribed in 1.5.b and the normalized Haar measure is a normal Levy family with constants
6.5.1.
Cl
= V1T/8,C2 = 1/8.
6.5.2. Similarly for each m the family X n = sn the product measure and the metric
X
sn
X •••
Sn(m - times), n = 1,2, ... , with
m
d(x, y)
=
(L p(x;, y;)2)1/2, X = (Xl,""
X m ),
Y
=
(Yl,"" Ym) E X n
i=l
(p-the geodesic metric in sn), is a normal Levy family with constants 6.6.
Cl
= V1T/8,
C2 = 1/2.
Next we show that homogeneous spaces of SOn inherit the property of being Levy
family. Let G be a subgroup of SOn and let V = SOn/G. Let II- be the Haar measure on V and let d n be the metric introduced in 1.5.d, Le. dn(t,s)
= inf{p(g,h);tpg = t,tph =
s}
where tp is the quotient map. Clearly II-(A ~ V) = lI-(tp-l(A) ~ SOn). By the definition of d n , tp-l(A~) :2 (tp-lAk Therefore, if II-(A ~ V) ~ 1/2, then lI-(tp-l(A) ~ SOn) ~ 1/2 and II-(A~) ~ II-((tp-l A)~). We conclude THEOREM: Let, for n = 1,2, ... , G n be a subgroup of SOn with the metric described above and with the normalized Haar measure II-n. Then (SOn/Gn,dn,lI-n),n normal Levy family with constants
6.7.
cl
= y'7rf8
and C2
=
1,2, ... ,18 a
= 1/8.
Theorem 6.6. together with Examples 1.5.b and 1.5.c implies immediately that the
following families are normal Levy families with constants
Cl
=
y'7rf8,
C2
6.7.1. Any family of Stiefel manifolds {Wn,k n };:C=l with 1 ~ k n ~ n, n
= 1/8:
= 1,2, ... ,
6.7.2. Any family of Grassman manifolds {Gn'k n };:C=l with 1 ~ k n ~ n, n = 1,2, ....
6.7.3. Exercise: Define a natural metric and probability measure on Vn,k = {€ E Gn,k;X E S(€)} and show that it is a normal Levy family (here, as before, S( €) is the unit euclidean sphere of the subspace
€).
6.8. We consider in this section a compact connected riemannian manifold M with II- being the normalized riemannian volume element of M. Then the Laplacian -D.. on M has its spectrum
consisting of eigenvalues 0
= Ao < Ar(M)
~
A2(M) .... The first non zero eigenvalue Al may
be represented as the largest constant such that
30
for every "sufficiently smooth" function
I
fM I
on M such that
= O. We refer the reader to
[B.G.M.l for more details. For the reader who is not familiar or does not feel comfortable with the notions above, we give a somewhat more detailed explanation on a model situation. We hope that this degression will help the reader to develop some intuition. Let X be a smooth n-dimensional compact connected C 2-smooth submanifold without boundary of the euclidean space (IR N
,
I . I).
Then norm
I . I induces
a metric on X. We,
however, consider a different metric, called the length metric, p: for x, y E X, p( x, y) is the length (with respect to
1·1 of the shortest curve in X
joining x and y. The Lebesgue measure
in IR n induces a measure on X. We normalize it so that the measure of X equals 1 and denote the normalized measure by J.l. We now have a metric probability space (X, p, J.l) For every x E X let T" denote the tanget (n-dimensional) plane to X at x. We write T"
= x + ~"
where
~"
is the parallel plane through the origin. Similarly the normal plane (of
dimension N - n) is N" = x + C:" where C:" E IR N • The space of pairs
GN,N-n
is the orthogonal complement of ~" in
T(X) = {(x,y); x E X, yET,,} is called the tangent bundle of X. We shall also use the dual bundle
T*(X) = {(x,z); x E X, z E T; = x + ~;} where ~; is the dual space to ~" (which we do not identify with ~,,). For e > 0 let U~(X) denote an
(1'1-)
e- neighbourhood of X in IR N
.
There exists an
e > 0 such that for every y E U~(X) there exists a unique x E X with yEN" (see [Mil). Therefore, given any I E C(X) we may extend it to j E C(U~(X)) by j(y) = I(x) for yEN". If
I-
is smooth enough we may consider
that the derivatives of
j
111-
N
(where standardly in IR ,11
N = Li=l Ix:). 2
Note
in the directions of N" are zero and therefore 11 may be written in sn ~ IR n +1, then, rewritting
terms of derivatives in the direction of T". For example, if X = 11 in spherical coordinates, we obtain the Laplacian on
sn,
(see [Vi], p. 493).
It is sometimes more usuful to see the Laplacian as constructed in two steps. In the
I on X the "il I(x) = "il II" E ~".
first step we introduce the gradient operation "il. For every Lipschitz function gradient is defined for almost all x and for such x is an element of ~'"
Indeed, it is more convenient to work with the dual space ~; and to consider "il I(x) as a linear functional (an element of G) we denote it then by dl(x)(E ~;) of course Idl(x)I' = l"ill(x)l. Then dl: X
-+
T'(X) or alternatively dl: T(X)
-+
IR by dl(x,y) = (dl(x),y) for y E ~". The
operator d from a dense subspace of L 2 (X) into L 2 (X
-+
T·(X),J.l) (the space of vector valued
functions F:X -+ T'(X) with IIFII = (J(IF(x)I;)2dJ.l)1/2,1·I;being the norm in ~;) can be extended to become a closed operator (also denoted by d)with the domain D(d). Let d' be the adjoint operator, then the Laplacian is given by
31 By construction, -A is non-negative. It is known that for a compact connected manifold
-A has a discrete spectrum, {A;}~O' Ai ~ 0, and that -AI = 0 only for I == const. It follows that AO = 0 and AI> O. The min-max principle implies now the inequality (*). 6.9. THEOREM: Let (M, p, J.L) be a compact connected riemannian manilold (J.L the normal-
ized riemannian volume). Let A
~
M with a = J.L(A) > O. Then, for all e > 0,
PROOF: Let A,B be two open subsets of M, J.L(A)
= a,J.L(B) = band p(A,B) = p > O.
Consider the function
I(x) = II a - (II p)(lla + 1/b)min(p(x, A), p), Let
fM IdJ.L
= a and apply
x E M.
(*) of 6.8. to get
Now, I is a constant on each of A and B so that the integration is on a set of measure
= 1 - a-b. I is a Lipschitz function with Lipchitz IV'II ~ (l/p)(l/ a + lib). Consequently we get
constant ~ (1 I p)( 1I a
+ lib)
so that
On the other hand
A1111 - aliL ~ Ad{ (f -
a) 2 dJ.L +
L
(f - a)2dJ.L)
= Ad(l/a - a)2 . a + (lib
~ A1(1/ a
+ a)2 . b)
+ lib).
Therefore,
1-a-b 2 AI' p ~ (l/ a + 1/b)(1 - a - b) ~ a. b or
b<
I-a 2
- 1 + AlP a
•
Fix a 8 > 0 and consider the sequence of pairs (Ai, Bi), i
= 0,1, ... , of subsets
defined inductively by:
A o = A, Bo =" ((Ao)s)C Ai+! Let ai
= J.L(Ai),
bi
= (Ai)s,
= J.L(Bi),
Bi+! = ((Ai+ds)C,
then
1-ab- < • • - 1 + Al .8 2 • ai
i = 0,1 ....
of M
32
Since ai
~
a for all
t,
bi
=
1 - ai+l ~
1+
1 - ai A 82 l'
.
a
Take 8 = 1/";;:;, then
l - a i+l
1 - ai
< -- l+a
and, by induction I-a 1 - ai ~ (1 + a)i'
If E: = i . 8 for some i, then i = E:~ and
1- J.L(A~)
= 1- ai
~ (1 - a)exp(-nj5:;log(1
+ a)).
< (i + 1)8. Then
In the general case pick the i such that i8 ~ E:
1 - J.L(A~) ~ (1- J.L(Ai.6)) ~ (1- a)exp(-E:~ log(1
+ a) + log(1 + a))
~ (1- a2)exp(-E:~log(1+ a)). EXAMPLES: 1. Al (sn) = n (this is the so called Wirtinger's inequality see [Gr2] and [Vi] p. 494) and we get: if A
<:::;
sn, J.L(A) ~ 1/2 then J.L(A~) ~ l-cle-c,~.;n. This is somewhat
weaker than what we got in 2.2.
2.
Let 1r
n
=
II?=l Sl then Al(1r n )
=
1 (on 1r 1 the eigenfunctions of -6. are
cos kx, sin kx, k = 0,1, ... with eigenvalues {k }%,,=o' For 1r n the eigenfunctions are prod2
uct of functions of the form cos kx, sin kx in the different variables; clearly one cannot get a positive eigenvalue smaller than 1). Here we get J.L(A) = 1/2 => J.L(A~) ~ 1 - clexp(-C2E:). Again, a better result may be obtained using 6.5.2., however not in a straight forward way. More precisely, There exist constants
Cl, C2
> 0 such that for every Borel subset A of 1r n
Note that in this example diam yn
~
yfii and so that E: = 8yfii is small with respect to
the diameter and for such E: we get a significant estimate.
7. MARTINGALES This section contains applications of some martingale inequalities to the local theory of Banach spaces. We begin with some elementary definitions.
9 be a sub a-algebra of 1 and let IE L 1 (0,1, P). 9, defines a measure on 9 which is absolutely continuous with
7.1. Let (0,1, P) be a probability space, let Then JL(A) = JAldP,A E respect to P19.
Consequently, by the Radon- Nikodym Theorem, there exists a unique
hEL 1 (0,9,P) such that JAhdP = JAldP for all A E 9. We call this h the conditional expectation of I with respect to 9 and denote h = E(f19). The operator I -+ E(f19) is easily seen to be a linear positive operator of norm one on all the L p spaces, 1 :::; P :::; 00. 7.2. Some additional properties of the conditional expectation operator are: 7.2.1. if
9'
is a sub a-algebra of
9 then E(E(f19)19')
7.2.2. if 9 E Loo(O, 9, P) then E(f· 7.2.3. if
9
is the trivial a-algebra,
g19)
9=
=
=
E(f19').
g. E(f19)·
{
E(f19)
= EI =
J
IdP.
7.3. Given a sequence 11 <;:; 1 2 <;:; ••• <;:; 1 of a-algebras, a sequence !J, h, ... of functions /; E LdO, 1;, P) is said to be a martingale with respect to {Ji}~1 if E(fiIJi-d = li-I for i = 2,3, ... A reader who feels uncomfortable with these definitions is advised to look at the following special case which is typical for most of the applications. 0 is a finite set, P is the normalized counting measure P(A) = IAI/IOI. {Oi}f=1 is a sequence of partitions of 0 each of which refines the previous one. Ji is the algebra generated by Oi. For a function f on 0, E(fIJi) is simply the function which is constant on atoms of Oi, the constant on each atom is the averaged value of 7.4.
I
on this atom.
All the martingale inequalities in these notes have a common feature: they evaluate
from above the probability of a large deviation of a function from its expectation.
Let I E L oo (O,1,P)'{4>,O} = 10 <;:; 1 1 <;:; ... <;:; 1n = 1, and put di = E(fIJi) - E(fIJi-d, i = 1, ... n. Then, for all c 2: 0, P(II - Ell 2: c) :::; 2exp( -c 2 / 4 2:7=1 Iidill~')· LEMMA:
PROOF: Notice that {E(fIJi)}i=o is a martingale with respect to {F;}f=o so that E(diIJi-d = 0, i = 1, ... n. Using the numerical inequality eX :::; x
+ ex2 ,
x E IR
34 we get, for all >. E IR,
E(eAdiIJi_d ::; E(>.diIJi-d + E(eA2d~IJi_1)::; eA2I1ddl~. It follows from the properties of conditional expectation that for all 1 :S i :S n, i
i
Eexp(>. L dj) j=l
i-I
= E(E(exp(>. L dj )IJi-d) = E(exp(>. L dj) . E(exp>.diIJi-d j=l
j=l
i-I
:S E exp(>. L
dj) . exp(>.2I1dill~).
j=l Now, for all >. > 0, n
P(/ - EI ~ c)
= P(Ldj
n
~ c)
= P(exp(>. Ldj - >.c)
j=l n
n
:S E exp(>. Ldj - >'c):S exp(>.2. L
j=l Putting>.
= c/
~ 1)
j=l Ildjll~
- >.c).
j=l
2Lj=1I1djll~, we get n
P(/ - EI ~ c) :S exp(-c 2/4 L IIdjll~)· j=l Similarly, n
P(EI - I ~ c) :S exp(-c 2/4 L Ildjll~) j=l and n
P(II - Ell ~ c) :S P(/ - EI ~ c)
+ P(EI - I
~ c) :S 2exp( _c 2 /4 L IIdjll~). j=l
o 7.5. To illustrate the way one uses the lemma, we will prove the following theorem, already stated in 6.4. THEOREM: The family TIn of permutation of {I, ... n} with the metric d(ll", e) = ~1{ijll"(i) t= e(i)} I and the uniform measure P (assigning mass tion) is a normal Levy family with constants C1 = 2 and C2 = 1/64.
;h
to each permuta-
Later on we will state a more general theorem (7.8.) 7.6.
PROOF: Let 1j, 0 ::; j ::; n, be the algebra of subsets of TIn generated by the atoms
{Ai, ,.. o,ij j 1:S i 1, .. . , i j :S n distinct} where Ail ,o .. ,ij
Then {
'0
~
' 1
~
•••
~ 1n
=
2
TIn
•
= {ll"j 1l"(1) = ill'"
,1l"(j)
= ij}.
This sequence of algebras has the following
35
Given any j, an atom of 7;. A = Ai.,...,ij and two atoms of Fi+1, B = Ai" ...,ij,r and C = Ai. ,... ,ij ,. contained in A; one can find a one to one map ep: B -> C
(*)
such that d(b,ep(b)) ~ ~ for all bE B.
Assume (*). If I is a function on II n with Lipschitz constant 1 and (li)'1=o is the martingale generated by I (i.e. Ii = E(lI7;)) then, for any A,B,C as in (*), li+1 is a constant on Band C and Ili+lIB - IH11cI ~ ~. Indeed, IH11B = li+lIC
= rbr L>rEC 1(7r) = TiT L>rEB l(ep(7r)) !/i+lI B - IH 1 1c I~
I~I
L
TiT L>rEB f(7r)
and
so that
1/(7r) - I (ep (7r))I
>rEB
~ l~l
L
d(7r,ep(7r))
>rEB
ItfollowsthatforanyA,Basin(*) Ili+lIB-liIAI ~~. Indeed, lilA
~ ~.
= n~i LC~Ali+1IC
so that Ili+lIB - lilAI ~ n~i LC~A Ili+11B - IH11cI ~ ~. Since this holds for all A, B as in (*) we get that Ildi+llloo ~ ~ for i = 0, ... , n - l. Applying the lemma we get that if I is a function with Lipschitz constant 1 then
(**) If A ~ II n then d(·, A) is a Lipschitz function with constant one so we can use (**). Choose c such that 2exp(-c::) =~, i.e., c = 4..)(log 4)/n. Then 1
P(ld(·,A) - Ed(·,A)1 < 4..)(log 4)/n) . > -2 On the other hand if P(A) ~ ~ then P(d(·,A) = 0) ~
!.
So there exists a 7r E II n such that
Id(1r,A) - Ed(·,A)1 < 4..)(log 4)/n and d(7r,A)
= 0,
i.e., Ed(·,A) < 4..)(log 4)/n.
Plugging this back into (**) we get
P(d(., A) ~ c + 4..)(log 4) /n ) ~ 2 exp( -c 2 n/16) or, for all
E:
> S..)(log 4)/n,
It is easily checked that this holds also for
E:
~
S..)(log 4)/n.
To prove (*) let p be the permutation which changes
T
with s and leaves the rest at
= po 1r then ep(7r)(i) = 7r(i) for all i except possibly for i = j + 1 (where 1r(i) = T,ep(1r)(i) = s) and for i = 7r- 1 (s) (where 7r(i) = sand ep(7r)(i) = po 7r(i) = p(s) = r), place. Define ep(1r)
so d(7r,ep(1r)) ~ ~.
0
36
7.7. The property (*) of II n suggests the following definition: DEFINITION: Given a finite metric space (n, d). We say that (n, d) is of length at most
£ if there exist positive numbers a1,' .. ,an with £ = (I:7=1 an 1/2 and a sequence {nk}~=o' nk = {An:,\, of partitions of n with the following four properties: 7.7.1.
mo = 1 i.e. nO = {n}
7.7.2.
mn =
7.7.3.
n
7.7.4.
for all k = 1, ... ,n,r = 1, ... ,mk-1 and i,j such that A~,Aj ~ A~-l there exists a
k
Inl
i.e.
nn =
{{X}}XEIl
is a refinement ofn k- 1,k = 1, ... ,n
one to one and onto function
->
Aj with d(x,
Note that £ is always smaller or equal to the diameter of
:S ak.
n.
The proof of the theorem below
is a direct generalization of the proof of the previous theorem and we leave it to the reader. 7.8.
THEOREM:
Let (n, d) be a finite metric space of length at most £, let P be the
normalized counting measure.
(i) Let f: n
->
R be a function satisfying If(x) - f(y) I :S d(x, y) for all x, yEn. Then for all
c~o
P{lf - Efl (ii) Let B ~ n, P(B) ~
7.9.
c2
~ c}:S 2 exp(- 4£2)
!' then for all c ~ 0
Another family of probability metric spaces which can be shown to be a normal Levy
family by the same approach is the sequence E!J: = {-I, l}n, n = 1,2, ..., with the normalized Hamming metric d((Ci)f=l' (oi)f=l) = ~I{i;ci # oi}1 and the normalized counting measure. With the natural choice of partitions one gets that the length of {-I, 1}n is at most 1/ Vii so that by 7.8. for all A ~ {o, l}n with P(A) ~
! and all c ~ 0,
This proves 6.2. (with different constants).
7.10.
REMARK: Note the difference in the order of deduction between this Chapter and
Chapter 2. There we used an inequality similar to the one in Theorem 7.8. (it) to deduce an inequality similar to the one in Theorem 7.8. (i). Here the order is reversed. As in Chapters 2 and 4 the inequalities we use in most applications to Banach space theory are for estimating large deviations of nice functions, like the inequalities in Lemma 7.4. and Theorem 7.8. (i).
7.11.
Our next theorem is a further abstractization of the method of the proof (7.6.) that
II n is a Levy family.
37 Given a compact metric group G with a translation invariant metric d (i.e. d(g, h)
=
d(rg,rh) = d(gr,hr) for all g,h,r E G) and a closed subgroup H. One can define a natural
metric d on G/ H by d(rH,sH) = d(r,sH) = d(s-lr,H).
The translation invariance of d implies that this is actually a metric and that d(r, sH) does not depend on the representative r of r H. 7.12. THEOREM: Let G be a group, compact with respect to a translation invariant metric d. Let G
= Go ;2 G l
;2 ... ;2 G n
= {I}
be a decreasing sequence 01 closed subgroups 01 G. Let
ak be the diameter 01 Gk-dGk,k = 1, ... ,n. Then
(i) II I:G c
-+
R is a lunction satislying I/(x) - l(y)l:S: d(x,y) lor all x,y E G, then lor all
> 0, n
JL(II - Ell ~ c)
:s: 2 exp( _c 2 /4 L
a%)
k=l
(il) II B ~ G with JL(B) ~
!' then, lor all c ~ 0, n
JL(B c ) ~ 1 - 2 exp( _c 2/16
L a%) k=l
(JL is the normalized Haar measure).
PROOF:
The implication (i) ~ (ii) is obtained as before using the function I(x)
d(x,B).
To prove (i), let l k , k
=
0,1, ... , n, be the a-algebra generated by the sets {gGd9EG.
Note that if gG k - l ;2 hGk then g-lh E Gk-l. If both gG k - l ;2 hlGk and gG k - l ;2 h 2G k , let s E G k - l be such that diam(Gk_dGk) = d(g-lhl,g-lh2Gk) = d(g-lh l ,g-lh 2s) and define rp:hlGk -+ h 2Gk by
Then
If follows that if we define
then the oscillation of
!k
on each atom of l k -
l
is at most ak, so that II d k II 00
:s: ak and using
Lemma 7.4. we get
o
38
'1.13. The two previous examples (II" and {-I, I}") are instances of this theorem. Another
example is 'lI''' with the normalized product measure I-' and the l1 metric
d(l,s)
" Lit; - s;l·
=
;=1 Taking 'lI'k, k
= 0, ... , n,
as subgroups, one gets ak :S 1 for all k. Then one can apply the
theorem to get e.g. that if B ~ 'lI''' with I-'(B) ~ ~ then
this should be compared with 6.5. where we used the euclidean distance. '1.14. We bring now two more applications of the method developed here. The first is a proof
of Khinchine's inequality used in 5.5. Recall that {T;}~1 are the Rademacher functions. THEOREM:
FOT
aliI :S p <
00
there exist constants 0 < A p , B p <
"
00
such that
"
"
A;1(L la;\2)1/2:S II La;T;llp:S Bp(L la;1 2)1/2 ;=1 ;=1 ;=1 for all {a;}?=1 ~
m.
Moreover Bpl..;p is bounded and A p is bounded away from zero.
PROOF: Fix {a;}?=1 ~ ill. with 2:~=1 a; = 1. {2::=1 a;T;}k=1 is a martingale (with respect to the algebras fk = the algebra generated by Tl,"" Tk)' Therefore,
so by Lemma 7.4., for all c
for 1 :S p
~
0,
< 00 we get, using integration by parts,
So, for 2 :S p
<
00,
The inequality :;;" :S et easily implies that SUPp~2 Bpl.,fP For p
= 1, let (J = i 1 = (fal
then ~
IL
= t+
<
00.
1~8 and, by Holder's inequality,
a;T;1 2)1/2 :S (fal
:S (fal
IL
a;T;1)8(fal
IL
I L a;T;1)8 . B~-8.
a;T;1 4){1-8)/4
39
Consequently,
Finally, for 1 :<::; p < 2,
o 7.15.
The second application is a different version of 5.3. in which one loses some of the
precision in the embedding of
e~n
into
er but gains some additional information on the form
of the embedding.
THEOREM: There exist constants 0 < a, b <
and 0 < a < 1 such that for all integers
00
k, n with k :<::; an one can find signs {ei; l;~l' ;:1 such that n
k
a(L a:J!/2
:<::;
i=1
n
aiXilll~
II L
:<::;
b(L a:J!/2
i=1
i=1
for all {ai} i~1 ~ JR, where 1
Xi
=-
n
Lei;e; E e~,
i
= 1, . .. ,k.
n ;=1
Fix
a = (ai) i~1 ~ IR with 2:~=1 a; = 1 and
let k
I(e)
= Ia:(e) = II Laixi(e)lll~. i=l
The triangle inequality implies that for any e,b E {_l,l}kn 1 k
I/(e) - l(b)1
n
L lail L lei; - bi;1 n
:<::; -
i=1
;=1
We want to make 1 into a Lipschitz function of constant one. So we define a distance function don {-I, l}kn by
40
Using the natural sequence of (kn) a-algebras, one checks that the length of ({-I,I}kn,d) is at most
V1/n,
so, by Theorem 7.8. (i)
P(lf - Efl
~
-c 2 n
c) :S 2 ex p (-4-)
P being the normalized counting measure on {-I, l}kn. Next we use Khinchine's inequality to evaluate Ef, 1
n
k
n
k
I LaiE"iil =
Ef = E- L
EI LaiTi(t)l.
i=l i=l
i=l
So
and we get that for all c
~
0,
Let now N be an O-net in Sk-1 with -1
P{A 1
-
INI :S exp¥-o
Then
_
c:S fa:S 1 + c, for all a E N} ~ 1 - 2 exp
2
(2k
c n 7i - ""4)'
If A 1 - c :S fa: :S 1 + c for all 0; E N. Then, for any 0; E Sk-1, we find by successive approximation {U;}~l ~ N and {Oi}~2 with 0 :S Oi < Oi-1 such that 00
0;
= U1
+ LOiUi' i=2
Consequently,
fa: :S (1
+ c)
f
= ~ ~ ~.
Oi
i=O
Also, for any
0;
E
sn-1, find Ii EN
with
10; -
iii < O.
> f-b - f-a-b- -> A 1- 1 -
~- ra
Then 1- c c- -0 1_ 0 '
and we get
P(A- 1 -c- I-cO < 1 1_ 0 Given
~_<
Ja -
1 +c for all 1- 0
a E Sk-l) > 1- 2 exp(2k -
0
e> 0 one can find c > 0 and 0 > 0 such that 1 +c - < 1 + ~ and Al-1 -
1-0"
C-
1 - Co A-I --0 > 1
1-
~
- ...
2
_ c n). 4
41 Then, for k
<
Hc:n -log2), 2k
1 - 2exp( -
8
c2 n >0 4
- -)
and one can find e: E {-1, 1}kn such that
o 7.16.
REMARK: It is known ([Sz1], [HaaJ) that Al =
Theorem b = 1 + €, a = ylfi tends to zero as € tends to zero.
€ for
any
€>
V2 so
that one can choose in the
O. In that case a seems to depend on
E and
8. EMBEDDING
e;
INTO
e~
This chapter is devoted to a more complicated application of the martingale method developed in Chapter 7. The proofs and the style here are more technical. The reader who is interested mainly in the outline of the local theory of normed spaces is advised to skip to the next chapter in first reading. In this chapter we shall estimate the dimension m of into
e~
e;, 1 < p < 2, which embed nicely
and see that m can be chosen to be proportional to n. This is the first application, in
these notes, of the method developed here to normed spaces which does not involve euclidean spaces. In Chapter 10 we shall see some more applications which do not involve euclidean spaces.
8.1.
p-Stable Random Variables. The proof of the main theorem in this chapter uses the notion of p-stable random variables.
Since these random variables are an important tool in the local theory of normed spaces, we take this opportunity to discuss some of their properties.
Proofs of facts about p-stable
variables which are not proved here (8.1.1 and 8.1.2 below) may be found e.g. in [Lo]. DEFINITION: A random variable g on a probability space (O,.1,P) is called symmetric p-stable, for some 0 < p
~
2, if Ee itg
for some c
> 0 and every
-00
=
1
eitog(W)dP(w) = e- clW
< t < 00.
We shall need two facts about these random variables: There actually are such variables for each 0 < p ~ 2 (for p > 2 there are no such
8.1.1.
variables!) and
8.1.2. The tail distribution of a symmetric p-stable variable satisfies the inequality
P(lgl 2: t)
~
Ct- P
,
t
>0
,
with C depending on c and p only.
< p (but
It follows easily that a symmetric p-stable variable belongs to Lr(O) for all r not for r
= p).
p-stables can be used to isometrically embed
ep
into L r for 1 ~ r < p ~
00.
Indeed, if
gl, g2, •.. is a sequence of independent symmetric p-stables with the same distribution (i.e.,
the c in the Definition 8.1 is the same for all the gi-S) and {ai}~=l are scalars, then,
E ",,,
(itt.a;<;) ~ )j «,,(ita,,;) ~ )j «,,(-, Itt'la;I') ~ E
E «" (
-'Itt' ~ la;I') .
43 So that, if Ej=llajlP
= 1, then
Ej=lajgj and gl have the same distribution (we recall that
the distribution of a random variable X is determined by its characteristic function Ee itX ). Consequently, n
Lajgj
j=l
r
for all n and all scalars {aj}j=l' Le., the span of the {gi}~l in L r is isometric to lp. From this it follows (using some technical perturbation arguments) that for any e > 0 and k there exists an n = n(k,e,r,p) such that l~ contains a k dimensional subspace E with d(E,l~) ::; 1+e. The estimate one gets on n this way is very poor. We are going to show that
actually n can be chosen to be of order k. To avoid complicated expressions and side-tracked technical computations we restrict our attention only to the case r
=
1. We begin with a
technical approximation lemma permitting us to replace· the p-stables by discrete random variables.
8.2. Discretization of p-stable variables LEMMA: Let 1 < P < 2, e > o. Let 9 be a symmetric p-stable random variable on [0,1], endowed with the Lebesgue measure A, and assume Ilgl\1 = 1. Let g"(t) be the decreasing rearrangement of Igi (i.e., g" is decreasing and P(g" > t) = P(lgl > t) for all 0 < t < 00) and
= g"C£), i = 1, .. :,n. Then there exists an a = a(e,p) such that for all m,n E IN with an, if Yl, ... ,Ym is a sequence of independent, symmetric random variables with each of IYil, i = 1, ... , m, having the same distribution as that of
let ai m ::;
n
L aiX('~' .1.J ' i=l
Y=
then
(1 - e) for all scalars bl
, ... ,
~ Ibjl (
lip P
m
::;
)
LbjYj
j=l
bm. (XA is the indicator function of the set A).
PROOF: Draw a picture to check that
Ilg" - ylll
=
{I {lin 10 (g" - y)dA ::; 1 g"dA. 0
Let C = C (p) be such that
P(g" > t) Then
=
P(lgl > t) ::; C . t- P
•
44 and
JIgO -
Let
yld>' ~
l
l/n
o
ll/n
gOd>' ~ Clip.
0
t-l/Pd>. = Cl/p_P_n(l-p)/p . p-1
gl, ... ,gm be independent symmetric p-stable variables with Ilgj III Zj = Eaisign gjX{a,:$lgjl
Then,
Zl, •••
,Zm have the same joint distribution as
Consequently, for all scalars bl m
Lbj(gj - Zj)
, ••• ,
J=
Yl, ..•
,m. Put
1, ... ,m .
,Ym and
bn , m
~ clip.
~ 1 . n(l-p)/p. L P
j=l
~
= 1, J' = 1, ...
Ibj[
j=l
Clip. _p_. (m)(p_l)/p. ~ p-1 n ( L
lip
Ib.IP ]
]=1
Choose a such that Clip . ....l!. a(p-l)/p p-l
<
g.
)
Then
m
Lbj(gj - Zj)
~ g(ElbjIP)l/p
j=l
and m
m
Lbjzj
~ L
j=l
bjgj
+ IIEbj (gj - zj)11
j=l
Similarly,
o From this lemma it follows immediately that l;,~el~ provided n ~ mm. As we have said above and will show below, this estimate can be greatly improved.
8.3.
Weak lp norms A second ingredient in the proof of the main result of this chapter (Theorem 8.8) is
a martingale inequality (Lemma 8.4) which is stated in terms of the weak lp norm. For
0< P < 00 and Ii = (al,'" ,ak) define
li lillp,oo =
max a~ . i l/p
l:$i:$k •
45
where
(ai)f=l is the decreasing rearrangement of (Iai I)f=l' The weak lp norm 11·llp,oo, satisfies al,"" at are vectors in IRk such that at
an "upper p-estimate for disjoint vectors", Le. if
each coordinate at most one of them is non-zero, then P
taill
Il
1=1
:S p,oo
t lIaill~,oo
.
1=1
a = Ef=l aiei, 7J = Ef=l13iei laio IP . jo then laio I is the jo largest among {Iail} U {113il}. Assume laio I is the i-th largest among {Iail}· Then lIall~,oo ~ laio IP . i, and among {113il} there are at least jo - i members larger than or equal to lail· So 117J11~,00 ~ laio IP. (jo - i) and Indeed, it is clearly enough to prove this for two such vectors
(lad 1\ l13il =
8.4.
0). If
lI a + 7J1I~,00
=
We return now to the martingale technique used in Chapter 7. The next lemma is a
consequence of Lemma 7.4. LEMMA: Let (0,.1, P) be a probability space and let
be a sequence of u-algebras. Let
f
E Loo(O,
1, P) and put
di = E(fIJi) - E(fIJi-l) , Then, for all 1 < P < 2 and all c
where'! P
+ .!q = 1
and S
P
i = 1, ... , k .
> 0,
= 8p(q+l)' (2-p) .
PROOF: Assume, without loss of generality, that
and choose a permutation
1r
of {1, ~ .. , n} so that
IId"'(k) 1100 Thus, we have for k
= 1,2, ... ,n,
Given an integer N :S n we have:
= Ildkll*,
k
= 1,2, ... , n
.
46
IE
But
d"'(k) \
~
E
II d "'(k) 1100
~ ~ k- l / p < qNl/q
so we get, by Lemma 7.4, P (
" IL
k=l
dkl
l
2: (q + 1)N /q
)
~
-N2/q [ 2 exp 4 E" lid k=N+l
~ 2 exp [
N 2 /q(1-
,..(k)
11 2 00
]
2/P)]
4N(1-2/p)
- P)N] = 2 exp [ -(2 4p If t 2: q + 1, set
[(_t )q]
N =
q+1
so that 1
< N < (_t _)q < 2N . -
-
q+1
-
Then
If t
~
q + 1, then
-(2 - P)t q ]
2 exp [ ( ) 8p q + 1 q
>2
-
exp
[-(2 -
p)]
8p
2: 2 e- l / 8 > 1 ,
and we get the desired inequality trivially in this case. D
8.5. Consider now the probability space 0 (e,1f) where
e is a matrix
= {-1, 1 }m
o "
x II;:', Le., the space of all couples
{ei,j}i~\,j~l with ±1 entries, and 1f is a vector (1rl, ... ,1rm) of
permutations of {1, ... , n}, with the normalized counting measure P(e,1f) = 2monl("I)m for all (l,1f) E 0. Fix 1 < P < 2, e > 0 and let m ~ an (a from Lemma 6.2). For each (e,1f) EO we define a sequence
Xl
(e, 1f), ... , Xm(l, 1f) of vectors in xi(e,1f)
=~L 1 "
f.~
by
ei,jaje",(j),
i
= 1, ... , m
j=l
(aj were defined above, (ej )j'=l is the unit vector basis in f.~). Our purpose is to find (l,1f) E
such that
n
47 for all scalars b1 , ••• , bm , with m as large as possible. We shall do it in two steps. First we show (Lemma 8.6 below) that for every fixed bi>" . , bm this is true on the average over
(g, w) E 0, then we show (Proposition 8.7) that for each b1 , • •• , bm
is very close to its average on most of 0. Now we use the new distributional inequality for martingales (Lemma 8.4). The estimates are such that they allow us to combine the estimates for different b1 , • •• , bm and still get good estimates for the deviation of II E~ 1 biXi(g, w) II from its mean, holding simultaneously for a lot of different (b 1 , •.• , bm
).
We choose the (b 1 , .•• , bm ) to form a o-net in the unit sphere of l; and conclude by approximating a general vector in the sphere of 8.6.
l;
by a member of this o-net.
We begin with an estimate of the mean.
m
l-E::;E :L.>jXj(g,w) j=1
PROOF: n
m
L L biEi,jaje..,(;) j=1 j=1
Fix 1 ::; k ::; n and consider the random variables Ui(g, w) = Ei, .., ' (k) .
a..,
1
(k)'
U1, . •• , U m
on 0 defined by
i = 1, ... , m .
These variables are symmetric and independent (check) and have the same distribution as Y1,' .. , Ym
in Lemma 8.2. So, by Lemma 8.2, for each k = 1, ... , n, 1 - E ::;
Elf
biEi''''l (k)a",' (k)
I::; 1 + E
1=1
and the same estimates hold when taking the average over k.
o 8.7.
We are now ready for the main part of the proof. For later use we state the next
proposition for a general norm replacing
l~.
48
PROPOSITION: a1
2 ···2
20
an
p
< 2, m, n
positive integers,
II . II
a norm on JR n ,
= ~ Ei=l ci,jaje".,(j))
and let
Ii = (b 1 , ••• , b m )
satisfy Ez:,llbil P
= 1.
Let,
n,
Then, for all c
where ~P
<
and Xi(E,7f) as before (i.e., aj = g*(jJn) for g an L 1 normalized p-stable
variable and Xi(E, 7f) for (E,7f) E
Let 1
>0
>0
depends only on p,
PROOF: Introduce the lexicographic order on 0 U {(i,j)}i~l,j~l' i.e.,
0< (1,1) < (1,2) < ... < (l,n) < (2,1) < ... < (2,n) < (3,1) < ... Let J o = {
{Jt : tEO U
{(i')')}i~l,j~l} is an increasing sequence of a-algebras the first of which is the
n.
trivial one and the last is the collection of all subsets of
For 1 ::;
i::;
m, 1 ::; )' ::; n let (i,)')'
be the immediate predecessor of (i,)') and define
In order to prove the proposition it is enough, by Lemma 8.4, to show that for all 1 ::; i ::; m,
1::; j::; n,
with K p depending only on p. Fix 1 ::; i ::; m, 1 ::; )' ::; n and an atom A of J(i,j)I, The value of EUIJ(i,j)') on A is the averaged value of EUIJ(i,j))IB where B ranges over all atoms of J(i,j) contained in A, and, for each such B, EUIJ(i,j))IB is the averaged value of
Ildi,jlB 1100
::; sup I C
f on B, Thus, for each such B
Av f(w) -
wEB
Av f(w)1
wEC
where the sup is taken over all atoms C of J(i,j) contained in A. Fix C, B atoms of J(i,j) contained in A. Then the values of
cu,v
and 1r u (v) are specified
and equal on Band C (and A)for all (u,v) < (i,j). Ci,j and 1ri(j) are also specified on Band on C but may be different, say, on B
Ci,j = CB,
1ri(j) =
Ci,j = CC,
1ri()') = t on C
S
while
49
Define a one to one and onto map (E, w) e*
u,v
=
->
(E*, w*) from B to C by
eu v
if
(u,v) =f (i,j)
ec
if
(u,v) = (i,j)
'
{
1r~= {
if
u
1r
l'f
P°1r u
u=fi u=t
where p replaces s with t and leaves the rest in place. Then
If(E, w) - f(E*, w*)1 = =
~ II~ ~
~ II~
bueu,vave".u(v) - bu<,vave".:(v)
biav[ei,v e "., (v) -
I
the two expressions in the parenthesis are equal except possibly when v = J' and when
v
= 1ril(t) so, by the triangle inequality,
Notice that
1r;1
(t) 2 j 'so that
la".;-, (t) I :::;
lai I and
If(E,W) - f(E*,W*)I:::;
4
-Ibillail max Ilekll . n l~k~n
It follows that,
I wEB Av
f(w) -
Av f(w)l:::;
wEC
4
-Ibillail max Ilekll n l~k~n
and
By the estimate on the tail of the distribution of a p-stable (see the proof of Lemma 8.2) we get
.
la")
=
n
and
By the upper p-estimate property of
lip
g*(!"') < c-I/p~
l!'l!p,oo
-
pip
50
and we get
11{ll d- -II ',]
00
}-~
-~
,-1,]-1
II p,oo -< 4C- 1/ Pn(l/p)-1 l~k~n max Ilekll = 4C- 1/Pn- 1/ q l~k~n max Ilekll . o
8.8.
Let 1 < P < 2, 0 < E:. Then there exists an {3 = {3(E:,p) such that if
THEOREM:
m ~ {3n then there exists a subspace X ~ l'1, dim X = m, and d(X,l~) ~ 1 + E:. PROOF: With notations as above we get, from Proposition 8.7 and Le=a 8.6 (here {ek}~=1 are the unit vector basis of
ll)'
h
~ 1
P(l - E: - c ~
+ E: + c)
q
~ 1 - 2 exp( -~p . c • n)
The rest of the proof is standard:
If N ~ S(l~), P(l - E: - c
I
~ II~ b;x;(e, 7f) ~ 1 + E: + c for all Ii E N) ~ 1 -
The right hand side is positive as long as
INI <
2lNlexp(
-~p . c
q
.
n) .
iexp(~p . cq • n) so that N can be chosen
to be a 6-net in the sphere of l~ if m ~ {3( 6, p, c) . n. If c
=
~
and 6 is chosen small with
respect to E: then
for all
Ii in N implies a similar inequality for all Ii in the sphere of
l~ (see for example the
proof of Le=a 4.1).
o
9. TYPE AND COTYPE OF NORMED SPACES, AND SOME SIMPLE RELATIONS WITH GEOMETRICAL PROPERTIES Let X be a normed space, Xi E X, i = 1,2, .... In this section we will study the averages ( Ave IIEf=1~ixiI12)l/2 and see that the order of magnitude of these averages gives a lot of e.=±l
information about some geometrical properties of X. We shall sometimes use a different way to write these averages
where ri are the Rademacher functions (see 5.5).
9.1. Given a normed space X, a natural number n, and 1 ~ p Tp(X,n) (resp. Cq(X,n)) be the smallest T (resp. C) such that
for all
Xl,'"
,X n
Let Tp(X)
~
2 (resp. 2
~
q
< (0), let
E X.
= sUPn Tp(X, n), Cq(X) = sUPn Cq(X, n).
If Tp(X) <
00
(resp. Cq(X) < (0)
we say that X has type p and/or that the type p constant of X is Tp(X) (resp. X has cotype q constant Cq(X)). When there is no confusion about the X we are dealing with we shall omit the symbol X. We have seen before (5.5) that for a Hilbert space H, T 2(H) 9.2.
Kahane's inequality. For any 1
~
p
= C2(H) = 1.
< 00 there exists a constant K p such that
This inequality seems to be closely related to the fact that
E'2 = {-I, I} n
is a Levy
family. However, we are not aware of a proof along these lines. A proof of inequality 9.2 is given in Appendix IU. 9.3. EXAMPLES:
1. L p , 1 ~ P ~ 2, has type p and cotype 2.
PROOF. We use the notation :::::l when the expressions on the two sides of sign are equivalent up to a constant depending on p alone (in this proof - the constants from Khinchine's inequality or Kahane's inequality).
52 For Xi E Lp(n, JL), i
= 1, ... , n,
(For further use, note that up to now the equivalence holds for all 1 ::; p Now, since
<
00.)
II . lie, ::; II . lie.
which proves that L p has type p. To prove cotype 2, use the triangle inequality (for integrals rather than sums) for the norm in l2/p
r/ ~ (~(1IXi(W)IPdJL)2/P r/ (~IIXiI12 r/
10 (~X;(W)
r/
2
dJL = 1
(~IXi(W)IP.(2/P) 2
2
dJL
2
=
o 2. L q , 2::; q <
00
has type 2 and cotype q. We leave the proof for the reader.
9.4. One may consider averages with respect to random variables other than the Rademacher
functions. Of particular interest are the gaussian averages. We recall that a normalized in L 2 gaussian variable is a random variable g(w) whose distribution is given by 1 P(g(w) ::; t) = . rr>= y
Let
{g;}~1
271"
Jt ,/2ds . e- s
-00
be a sequence of independent gaussian random variables normalized in L 2 •
For 1 ::; p ::; 2 ::; q <
00
and n E N we define the gaussian type p (resp. cotype q) constants
O:p(X,n) (resp. ,8q(X,n)) of X as the smallest T (resp. C) such that
(Ell ~ 9i(W) Xi 2) 1/2.::; T (~IIXiIIP) I1
(",p (t, 1Ix;II'f
I/p
~ c (Ell t, ,,(wlx;II'
r)
53
) for all
X1, ••• ,X n
E X.
We want to compare lrp(X, n) and .Bq(X, n) with Tp(X, n) and Cq(X, n). First note that the distribution of gl (w), ... , gn(w) is the same as that of T;(t) Ig;(w) I, ... , Tn(t) Ign(w) I, so, by convexity, (
Ell
t; g;(w)x;11 2 n
)
1/2
t; T;(t)lg;(w)I~;1l2 ~ Etll t; T;(t)E lg;(w)lx;1I 2 V2Ti. Ell t; T;(t)x;11 (
=
Ell
n
)
) 1/2
n
w
(
n
=
1/2
2
) 1/2
(
and we get
9.4.1. and
9.4.2. Next, notice that
so that
9.4.3. Using Kahane's inequality one gets similarly that, for all 1 :::; P :::; 2,
9.4.4. Also, a similar computation gives,
9.4.5. and
9.4.6. (which is better than .Bq(X,n):::; V7r/2. Cq(X,n) for q
= 2 and other values of q close to 2).
54 The remaining inequality, namely Cq(X, n) ::; K q . ,6q(X, n) does not hold in general. We discuss this matter in Appendix II.
9.5. The next simple computation shows the relation between the gaussian average and another average we have previously used: integration with respect to J1.- the Haar measure on sn-l.
C n and C~ depend only on n so one can compute them using (xi)f=l an orthonormal system
in a Hilbert space and get
Le.
9.5.1.
We now bring two theorems relating the notions of type and cotype to geometry. 9.6. THEOREM: Let X be a n-dimensional normed space with cotype q constant C q for some q < 00. Then k(X) ~ c· C q 2 • n 2 / q for some absolute constant c. (k(X) was defined in ,4.9).
PROOF: Let
I. I be the euclidean norm on X
given by the ellipsoid of maximal volume
contained in B(X). Then, by Theorem 3.4, there exists an orthonormal basis {ei}f=l with
Ileill
~
1/4 for 1::; i ::; n/2. Therefore (by Theorems 5.1 and 3.3) Mil_II
~ ( 1r
IIEaieiI12dJ1.(a)) 1/2
5 ,.,-1
=
(In-l folll~airi(t)eiIl2dt dJ1.(a)) 1/2
~ (2. Cq)-l
[n/2]
(
In-l (~ lail q
)
q
1/2
)2/ dJ1.(a)
~ (2 . Cq)-1(n/2)~-~ ( Jrsn-l 10 ~ la;\2dJ1.(a) J
= (2.
Cq)-l
Cn~21) 1/2 . n~-!
)
1/2
55 and by Theorem 4.2, k(X)
>
-
C1 •
2 . n 2/ q n . M r2 > c . Cq
•
o REMARK: The proof actually shows that the factor C q can be replaced by Cq(X,n). 9.7. THEOREM: Let X be a normed space whose gaussian type 2 constant on n vectors is
= o:(X,n).
o:(n)
Then k(X) ~ c . 0:(n)2
for some absolute constant c.
PROOF: We actually prove a somewhat stronger theorem: If
for some {Xj}i=l ~ X, not all of which are zero, then k(span{x;}f=l) ~ c· For
0: 2 •
Fix m, k E :IN and consider the probability space {-I, 1}k.m.n with the uniform measure. E {-I, l}k.m.n define
"t
1
u,("t)
n
k
= Yk L
L e,,j,txj j=l t=l
and, given independent, normalized gaussian variables {g"j} ,~\,j~l' define k
v,(w)
= Lg"j(w)Xj
.
j=l
Note that the central limit theorem implies that the distribution of (U1, ... ,um) tends, as k
-> 00,
Let
to that of (VI,'" ,v m ).
a = (a1,"" am)
be such that E~ 1 a:
= 1 and consider the function m
f("t)
= fa:("t) = II L
,=1
a,u,("t) II
.
We form the obvious sequence of k· m' n (f fields fs,j,t (Le. an atom of fs,j,t is a set where all of eu,v,w are specified for all (u, V, w) ~ (i,j, l) in some pre-specified, say the lexicographic, order) and we get a martingale Is,j,t
= EUlfs,j,t)
with martingale differences (d"j,t}':'l,j~l,t~l' We leave it to the reader to check that
56
By Lemma 7.4 we get
Let k
-+ 00
to obtain
Note that since E~l a: = 1, (E~l aigi,l,"" E~l aigi,n) is again a sequence of independent gaussian variables normalized in L 2 so that n
m
Ell Laivi(')11 =EIILgjxjll.
i=l Choosing
C
= E:' EIIEj=lgixill
j=l
we get
The rest of the proof is standard (see the proof of Lemma 4.1).
o 9.8. Given a measure space (11, Y,p,) and a Banach space X, we define the space L 2 (X) = L 2 (X,I1) as the space of all measurable functions from (11, Y) to X with 1/2
IIIII =
(
<
111/(w)llidp,)
00 •
In the cases we shall be interested in, the space 11 will be finite and no problem of measurability will arise. Actually, the only measure space relevant to us is the space {-I, 1}n with p, - the normalized counting measure. For IE L 2(X), 9 E L2(X') we define
< g,1 >=
J g(w)(f(w))dp,.
relation. Dimension consideration actually shows that L 2 (X)* norm 9.9.
11'lli,(x) = II·IIL,(X·), when X
=
This defines a duality
L 2 (X') with equality of
is finite dimensional.
Given any orthonormal basis {W"'}"'EA for L 2 ({ -1, l}n,p,) one can expand any
IE L2(X) as I(t) = E"'EAW",(t)X", with some {X"'}"'EA ~ X. Indeed x'" = J I(t)w", (t)dp,(t) (note:this integral is actually a finite sum). We will be interested mostly in one particu-
lar basis:
ri(t)
the Walsh system. Let (ri(t))i=l be the Rademacher functions on {-I,I}n, Le.,
= ri(t 1 , ... ,tn ) = ti.
For A ~ {I, ... , n}, A
WA(t)
#- 0, let
= II ri(t) iEA
57 and let
It is easily checked that {WA}A~{I•... ,n} is an orthonormal system of 2 n functions in a
= W{i}'
space of dimension 2 n so it must be complete. Note that Ti
There are several other ways to introduce the Walsh system: Let define the 2 n x 2n matrix W n (called the Walsh matrix) by
Wo
= 1 and inductively
(for example 1
1
-1
1
1
-1
-1
-1
-1
1
-~) ,...
).
Then one can identify the points {-1, 1}n with (1, ... , 2n ) in such a manner that (WA(t))A~{I•... ,n} are identified with the 2 n rows of W n . Alternatively, {WA} A~{I, ... ,n} is the group of characters of the multiplicative group {-1,1}n. 9.10. We will be interested in one special subspace of L 2 (X, {-1, 1}m) - the one spanned by
the first n Rademacher functions: RadnX
=
{~Ti(t)Xi;
Xi EX, i
There are natural projections onto these subspaces; for f
= 1, ...
=
n} .
EA~{I •... ,m}WA·
XA we define
n
Radnf
=L
Ti . XU} .
i=1
The norms of these projections playa central role in the theory of type and cotype.
Let X be a normed space, n an integer and let 1 < p ::; 2 ::; q < Then
LEMMA: 1p
+ 1q = 1.
PROOF: For any g(t)
= E~ITi(t)xi,
xi EX',
00
with
58
Given Xi E X, i and
IIxiliP = Ilxillq.
= 1, ... , n, let xi EX',
i
= 1, ... ,n, be such that xi(x;) = Ilxili ·llxill
Then
n ) lip n n q) llq ( n ~ II xi liP = ~ Ilxill·llxili = ~ Xi(Xi) ( ~ Ilxill n
n
n
n
=< LTi(t)xi, LTi(t)Xi >:S II LTi(t)xi IIL,(XO) ·1/ LTi(t)XiIIL,(X) i=l
i=l
i=l
i=l
Therefore,
Thus,
To prove the right hand side inequality, note first that for any m
f(t) = LTi(t)Xi + L
WA(t)XA E L 2 (X) ,
IAI~l
i=l
n
IIf(t)IIL,(x) 2 II Rad nll- l ·11 LTi(t)xdIL,(X) i=l
2 IIRadnll- l . Cq(X, n)-l
(~IIXillq) llq
Given xi E X', i = 1, ... ,n, let m
f(t) = LTi(t)Xi + L i=l
WA(t)XA
E
L 2 (X)
IAI#
be such that IIfliL,(x) = 1 and n
n
II LTi(t)xiIIL,(X O) =< i=l
Then
LTi(t)xi, f(t) > ';=1
n
n
II LTi(t)xiIIL,(XO) =< i=l
LTi(t)xi, f(t) > i=l
n
=
( n
:S IIRadnll· Cq(X,n)
t; Ilxill q) llq
) lip (n
~ xi(Xi):S ~ Ilxill P
.
·llf(t)IIL,(X)·
= IIRadnll· Cq(X,n).
n ) lip ( ~ II xi liP
n ) lip ~ II xi liP (
59
Thus,
o 9.11. As an immediate corolJary we get that, if X' has type p, X has cotype q (~+
and that if type p.
SUPn
IIRadn11 <
The assumption type> 1.
SUPn
00
then also the converse holds:
IIRadn11 <
00
is realJy needed:
We shalJ return to the subject of estimating
IIRadn11
i
= 1) if X has cotype q then X' has
II has cotype 2 while loo has no
in Chapter 14.
CHAPTER 10: ADDITIONAL APPLICATIONS OF LEVY FAMILIES IN THE THEORY OF FINITE DIMENSIONAL NORMED SPACES
We saw in the previous chapters some applications of the concentration of measure phenomenon on sn. For example, we have proved in Chapters 4, 5 and 9 Dvoretzky's Theorem and some of its generalizations. We have also seen applications of the same phenomenon and of some martingale techniques in the families {En:;O=l and {I1 n }:;O=l (see e.g., Theorem 9.7 and most of Chapter 8). In this chapter we will see a somewhat different kind of applications using some other Levy families. Some of the examples will be described only briefly leaving most of the details to the reader.
10.1. Let (X, p, p,) be a compact metric space with a Borel probability measure p, and let f be a continuous function on X, f E C(X). As we have seen in Chapter 5, it is useful to compare the median Mj with the average E(f) = fd w To estimate the difference Mj - E(f) we use the same approach as in Lemma 5.1.
Ix
Let
ax(c)
= sup{1- p,(Ae)j
1
A S; X, p,(A) ~ Z},
Take any partition 0 = co < Cl ... < cN = diam X and put 0; = Wj(c;) where, as before, Wj(·) is the modulus of continuity of f. Then
IMj - E(f)1
~
N-l
L r IMj ;=0 JO;
fldp,
~
N-l
~
L 0i+l . p,{Xj IMj -
N-l
f(x)1 >
od
~2
;=0
For example, in the typical case where ax (c) ~ for any c >
1/..;Ti (choosing Cm
L Wj(ci+d . ax(c;) .
;=0
=
C'
Cl .
exp( -C2 . c 2 • n) and
W
j( c)
~
C . c we get,
m), 00
IMj - E(f) I ~ 2· C . c . Cl
.
L (m + 1)exp(
-C2 .
m 2 . c 2 . n) ~ K . C . c
m=O
where K = K(Cl,C2)'
10.2. Let (X, I ·11) be a normed space. A sequence {xdf=l in X \ {a} is said to be Kunconditional (resp. K-symmetric) if for al1 {a;}f=l E IRk and al1 c; = ±1, i = 1, ... ,k,
61 (resp. for all {adf=l E IRk, all ei = ±1, i = 1, ... ,k and all
'Tr
E Ilk
The importance of unconditional and symmetric sequences stems from the fact that a lot of the desired structure theory, which does not hold in general normed spaces, holds in spaces spanned by such sequences (see [L.T-I,1I1 and references there). 10.3. In our first application in this chapter we use the normal Levy family from 6.5.2:
X
=
Il~l S~) I where each of S~) I is the euclidean sphere sn-l.
p(x,y) = (E~IPi(Xi'Yi)2)1/2 (pi the geodesic metric on S~)l), for x y = (YI,"" Ym), and the measure JL is the product of the Haar measures. THEOREM:
The metric is =
(XI,''''X m ),
Let E be a (finite or infinite dimensional) Banach space with cotype q
constant Cq , for some q <
00.
Denote S(E)
= {x E Ej Ilxll = I}
ep : Sn-l
-+
and let
S(E)
be an antipodal map (i.e. ep( -x) = -ep(x)) with Lipschitz constant L (i.e. lIep(x) - ep(y) II ~ L·p(x,y)). Then, for each TJ > 0, there exists a sequence {Xi}~l ~ sn-l such that {ep(Xi)}~1 forms a (1 + TJ)-symmetric sequence in E and
where c(TJ) > 0 depends on TJ alone. If E has no cotype the same conclusion holds with q = (in that case q/(q - 1) PROOF: For a
00
= 1 and Cq = 1).
= (al,'"
,am) EIRm and x = (Xl, ... ,X m) E X (= Il~IS~)I), let
The function Illalll = E(Ja) on IR m is a norm. It is easily seen that the natural basis is I-symmetric in this norm. We are going to show, using the concentration of measure phenomenon on X, that, for some x, fa(x) is close to Illalll for all a E IR m • Then {Xi}~l will be the desired sequence. We devide the argument into several steps. 10.3.1.
Let M a be the median of fa on X and w a (-) its modulus of continuity. Then, by
Theorem 6.5.2,
Now,
62 and we get
t; !ail m
J.t{X;
Ifa(x) - Mal < L· e· m!-!.
10.3.2. Now we replace M a by E(fa)
(
= Illalll.
IJq
)
q
}
Since, by 10.1,
IMa -lllallll :S K· L· e· m!-~. ( for some universal constant K (as long as e
~ 1- ~ exp(-e 2 • n/2).
t; lail m
q
IJq
)
> 1/..;n), we get
10.3.3. Using the antipodal property of 'P and the cotype inequality we get,
Fix (J
> 0 and let e
= (J{(K + 1) . L· Cq}-l
1
1
. ml1-~. Then
and we get 10.3.4. J.t{Xj
lfa(x) -lllallll < (J • III aIII}
CJ;;
~ 1 - V 1r /2 exp
Now pick a 6-net N in the sphere of the norm
_(J2 • n
(
[m1-(2/q).
III . III .
2 . {(K
+ 1) . L. CqF]
One can do it with
(see Lemma 2.6). Therefore, if 10.3.5.
CJ;;
V 1r /2 exp
then one can find
xE
(1- (J)
(2m - 6
(J2 • m 1 -(2/q)
n
·2· {(K
+ 1) . L . Cq }
2
)<1 ,
X such that
·llIalll:S
II~ai'P(xi)11
It follows (as in 4.1 for example) that 1-1-6 26 - (J
·lllalll:S
:S (1 + (J) ·lllalll for all a EN.
I
II~ ~ai'P(xi) :S
1 + (J 1-6
·lllalll
INI :S
)
.
e2m / S
63 for all
a E mm.
+ 0)/(1 -
In particular, {
6 of the same order and such that (1
+ 0)/(1 - 0 -
26 - OJ-symmetric. Choosing 0 and
26) = 1 + T/ gives, using 10.3.5., the right
estimate for m.
o 10.4. For the next result which is similar in nature to the previous one, we need the notion of
a block sequence: Given a sequence
Xl, ..• , X n
of elements of some linear space, any sequence
of non- zero vectors of the form EiE<TiaiXi, j = 1, ... , m, with {(7i}~1 disjoint subsets of
{l, ... ,n}, is called a block sequence of XI,""X n , THEOREM: For any c > 0 there exists a constant c(c) > 0 such that, given any n and a
sequence Xl, .•. , Xn of linearly independent vectors in a normed space E, with cotype q constant C q, there exists a block sequence {YI."" Ym} of x}, ... , Xn which is (1 + c)-unconditional and m ~ c(c)nl/q/Cq.
PROOF: We give only the beginning of the proof leaving the rest to the reader. Divide
{l, ... ,n} into m almost equal parts find, using Theorem 9.6, a (1
{Ai}~l'
IAil
+ ~)-isomorphic copy
~
k ~ n/m. In each Ei = span{xihEAi
Fi of e~ for t ~ c(c)k 2 / q/C q. Define now
o 10.5.
Finite dimensional version of Krivine's Theorem. In Chapter 12 below we shall state
and prove a theorem of Krivine about embedability of
e; spaces in general Banach spaces.
Here we present a special case of this theorem with a proof which, unlike the proof of the general theorem given in Chapter 12, allows one to get reasonable estimates on the dimension of the embedded
e; space. lip be the ep norm n be another norm on m and assume
Let {e;}i=l be the unit vector basis in IR n and let II·
IIEi=laieillp
= (Ei=daiIP)I/P).
Let 11·11
on
mn
(i.e.,
THEOREM: For any c > 0 and C < 00 there is a constant a = a(c, C) such that for mn satisfying (*) there exists a block sequence UI ... ,U m of el ... , en which
any norm 11·11 on satisfies
forallal, ... ,a m
I (~/36C)P
andm~a'n3
.
64 The proof consists of two parts. In the first we find a symmetric block sequence g > 0 there exists a constant C(e) > 0 such that if II ·11 is a norm (*) then there exists a block sequence {Yl, ... ,yd of {el, ... ,e n } which is 21' (1 + e)- symmetric and k ~ C(e) . C-T . n l / 3.
10.6.
LEMMA: For any
on /Rn satisfying
PROOF: Consider the family {Xn = E'2 x IIn}~=l with the metric
= ~n I{i;
p{(t,7r), (s,ip)}
ti
#
Si aT 7r(i)
#
ip(i)}l.
Using the natural sequence of u-fields (an atom in the k-th u-field is specified by the values of the first k components of t E E'2 and the first k values, 7r( 1), ... , 7r (k), of 7r E lIn) it is easily seen that the length (see 7.7) of X n is S 2/.,fii. By Theorem 7.8 we get that X n is a normal Levy family (with. constants
Cl
= 2 and C2 = 1/64).
= k .m
(k to be chosen later). Divide {I, ... , n} into k disjoint sets AI, ... , Ak each of cardinality m. For (t, 7r) E X n let Without loss of generality, we may assume that n
Yj
= Yj(t, 7r) = m- l / p
L
tie ..(i),
J'
= 1, ... , k.
iEAj
For
a = (al"
.. ,ak) E IRk, consider the function k
fll(t,7r) =
L ajYj j=l
on X n • We now estimate the modulus of continuity of fll
Wa(e)
= m-
l p / •
sup {
t
aj
L
tie ..(i)
iEAj
j=l
k
L aj L (tie ..(i) j=l
sie'l'(i»)
iEAj
For each j the maximal number of i - s such that tie ..(i)
#
sie'l'(i) is e . n.
Thus, we get,
Clearly, E(fa) now yields
= IlIalll is a I-symmetric norm on IRk.
The same computation as in 10.3.2
65 Now
plugging this into the inequality above yields, as in 10.3.4,
1L{(t,1r); (I-(J)lllalll::;
k [ 2 ( 2C (J ) 2P] ~a;Y;(t,1r) ::;(I+(J)llIalll}2:1-2exp~: .
The rest of the proof follows steps 10.3.5 - 10.3.6 in the proof of Theorem 10.3 and, we hope, can be completed by the reader.
o 10.7.
When proving Theorem 10.5 we may now assume, using Lemma 10.6, that we are
dealing with a I-symmetric sequence. The next theorem will then complete the proof of Theorem 10.5. THEOREM: If Y1,' .. ,Yk is a l-symmetric sequence in a normed space satisfying k
C1
1
11 allp::; II LaiYil1 ::;
C 2 11 allp
i=1
for all a = (a1,"" ak) E IRk. Then, for all ~ > 0, there is a block sequence U1, . .. ,U m of Y1, ... ,Yk satisfying m
(1 - ~) ·llall p ::; II
L aiuill ::; (1 +~) .lIall p i=l
foralla=(a1, ....am)EIRm wherem 2:
r 3P k r ,
r=(~/36C)P, C=C 1 ·C2 •
PROOF: After nine chapters of results in which we found nice substructures of given structures, using probabilistic method, it is an interesting feature of the following proof that it is deterministic. One can actually write a formula (depending on
~,p
and k) for the block
sequence {Ui}~1' We begin with a description of this formula. Let
"~"
mean "is a rearrangement of" and "$" mean "sum with disjoint supports". Let
Yr stand for the ~ equivalence class of the sum of r distinct elements of {Y1," . ,Yk}. Let s = 1, ... ,m, be disjoint with
z.,
N-1
z. ~
L
$p(N-j)/p.
Yip;]
;=0 where p = (a
+ I)/a
and the integers a, N, depending on ~, p and k, will be chosen later.
The integer m is determined by
k~m'
N-l
L
j=O
[pi] ~m·
N-l
L
j=o
pi
$;mpN/(p_I) =ma«a+l)/a)N).
66
Finally we set U
8 = z8/llz811,
s
= 1, ... ,m.
The dependence of a and N on c, p and k may be traced through the following formulas which we bring in detail in order that the dependence of the different parameters in the proof will be clear:
co
= ~(c/2C)P, 6 = co' m- 1 / p , a = [1/6pJ = [m 1 / P/pco], (a/(a + 1))[N~oJ ~ c~/m. 12
The substitution of all the relations above in (*) will determine m. We now give a formal proof. Fix T < N, M ::; m and t s e {a, 1, ... , T}, s = 1, ... , M, with M
L:>-t
8
= 1 + 77,
1771::;
1.
8=1
Then M
N-l+T
L
Lp-t 8 / P Z8 ~ 8=1
S=l,
j=O
L
EEl p(N-j-t 8 )/PlI".ij ~ ,M
EEl p(N-i)/PYai ,
i=O
N-l
where ai
L
=
[pi- t 8 l·
.E{I .....M}
i-t.E{O•...• N-l}
Observe that (for ie{O, ... ,N-l})
[pi] _ ai
=
(f
p-to - TJ) [pi] - ai
8=1
=
L
i-t.
p-t. [pi]
+
L
([pi]p-t. _ [pi-t.]) _ TJ[pi].
i-t.~O
Comparing E~IP-t./Pz., with the appropriate rearrangement z of
ZI,
satisfies A ~ '"" _
N-1 ""
~
i=O
N-l+T
p(N-i}/Py;. I[p')-a.1 '"
"" ~ i=N
'" p(N-i}/py,a...
the difference /1
67 Hence
On the other hand,
Ilz.1I ~ (1/CI) pN/p(N/2)1/
P,
hence the relative difference is
111~p-t'/P'Usll-ll ~ 1I~lIjllz,,1I ~ C(8TjN +21771 +2Mp-T)1/
.
and this is ~ c/2, provided T :S Nco, assume, therefore, T = [Nco]. Let now
1771 :S co and Mp-T
~ co, where co =
P
,
f.J.(c/2C)P. We
We want to replace the as by some f3s = p-t./p , t s E {O, ... , T} or by f3s that
lIasl - f3sl
~ 0 'tis where 0
= eom- l / p .
~ c~/m and that 1 this. The choice of 0 implies that (a/(a
+ l))INeoJ
and that
p-l/p
=
For this we ask that p-T/p ~ 0,
~ 8. Taking a
= [l/op] = m 1/ p /pco
0, so Le.,
guarantees
68
1112::'=1 QstLsll- 11
< E. The conditions which have to be guaranteed are thus, oJ m :s; Eg((a + 1)la)!Ne and ma((a + 1)la)N :s; k, when a = [m 1/ PIEOP]. The substitution
Hence
(
a)
[Neol/p
--
a+l
yields m1/p
m-EOP
(m
1/
-1
~Eom
/p
p)p/eo
~k,
-EO
i.e., m ~ E~(l+eo/p)/(l+eo(1+1/p))(pk)eo/(l+eo(1+1/p))
> -
where
r = (E/36C)P
(_E_) 24C
(provided
E
5p2/2
k(e/36C)P
> -
r3Pkr
< 6C). o
PART II: TYPE AND COTYPE OF NORMED SPACES
11.
RAMSEY'S THEOREM WITH SOME APPLICATIONS TO NORMED
SPACES 11.1.
We begin with the statement and proof of Ramsey's Theorem. For a set B and a
positive integer k let Blkl denote the set of all subsets of B with k elements. THEOREM: Let A be a finite set, k a positive integer and let I: BVlkJ --> A be any function. Then there exists an infinite subset M ~ IN such that I(MlkJ) is a singleton. A simple compactness argument shows that the following finite version of the theorem also holds.
For all positive integers k, n, m there exists an N such that if I: {I, ... , N}lk] --> {I, ... , m}is any function then there exists a subset M ~ {I, ... , N} with IMI = n such that f(Mlkl) is a singleton.
11.2.
THEOREM:
The behaviour of N as a function of k,n,m is a much studied subject (cf. [G.R.S.]). 11.3. PROOF OF 11.1: We shall prove the Theorem by induction on k. The case k evident. Assume the Theorem holds for k and let subsets of IN of cardinality k function
1m. :
+ 1 to
(IN - {md)lkl
-->
I:
IN[k+l!
-->
= 1 is
Abe any function from the
a finite set A. Let ml E IN be arbitrary and consider the
A given by
By the induction hypothesis there exists an infinite M l
I( {ml, nl, ... , nk}) is a constant for all {nl ... nd
~
<;;; IN - {md such that
M l . Let m2 E M l and choose an infi-
nite M 2 ~ M l - {m2} such that I( {m2, nl,"" nk}) is a constant for all {nl,' .. , nk} ~ M 2. Continuing in the same manner we find a sequence of infinite subsets IN
= M o ;2 M l
;2 M 2 ;2
M3;2 ... of the integers and a sequence ml,m2,." with mi E Mi-l - Mi, i = 1,2, ... , and
for all {nl, ... ,nk} ~ Mi. Choose an infinite subsequence M ~ {ml,m2,"'} such that the constant in (*) is the
70
same for all the members of M. M is then the desired subsequence.
o 11.4. We turn now to the first application to normed spaces. THEOREM: Let {Xi}~l be a sequence in a normed space X with the property that for all n and all scalars al, ... , an not all (depending on
ab""
0/ which are zero, there exist constants 0 < c <
C
<
00
an) such that for all i l < ... < in n
c:S LajXij :SC. j=l Then given any sequence el > e2 > IN such that for all n, all m n < i l <
> 0 there exists an infinite subsequence M = {m n } ~ < in, m n < fl < ... < fn' ik,fk EM and all scalars
al,'" ,an
PROOF: We build infinite sequences N l ;2 N 2 ;2 ... of integers such that the assertion holds for N n for a fixed n. M will then be the diagonal sequence. To build N n given N n- l we proceed as follows:
Fix al,"" an, divide the interval
[c, C] into finitely many disjoint intervals of length at most S > 0 (to be chosen later) and let f: Ni~l be the function which assigns to ill"" in' satisfying i l < ... < in' the number of the interval to which II Ei=lajXij II belongs. By Ramsey's Theorem 11.1 one can find a subsequence M l such that IIEi=lajxijll always belongs to the same interval for il, ... ,in E MI' If S > 0 is small enough, this ensures that
for any two sequences i 1 < ... < in' fl < ... < fn in MI' We now repeat the process for a different sequence aI, ... , an to find a subsequence M 2 of M 1 good also for these scalars. We repeat the process finitely many times to take care of all scalars (al" .. , an) in a fine enough net of B(li). The last sequence will be N n . We leave the details to the reader.
o 11.5. COROLLARY: Let M be the sequence given in Theorem 11.4. Then the sequence Yi=Xm~i-Xm:2i_l
satisfies
,i=1,2, ...
71 for any finite sets A ~
B
with min
B 2 IBI.
PROOF: Fix a finite set B and scalars {ai};EB. Fix n E :IN and let {Ui};EB be subsets of M with the following properties: (i)
(ii)
if i > ;" then min Ui > max Uj min Ui
>
(iii)
for i E A
(iv)
for i E B
mIBI.(n+l) for any i E B
IUil = 2. Say, Ui = {J(i, I),J'(i,2)} - A IUil = n + 1. Say, Ui = {J(i, 1), ... ,;"(i,n + I)}.
By the property of M (almost invariance under spreading) we get for any 1
L
Il .EA
ai(Xj(i,2) - Xj(i,l))
ai(Xj(i,k+l) - Xj(i'k»)11
.EB-A
Averaging over 1
(1
+ .L
+ en) I L
'EB
~
k
~
aiYil12
~ lin L ai(Xj(i,2) -
Xj(i,l»)
'EA
.EA
-+ 00
IlL
k
~
n
aiYil1
'EB
n, we get
= II?: ai(Xj(i,2) Letting n
~ (1 + en)
~
Xj(i,l))
+
t .L
k=l
+ ~ .L
ai(Xj(i,k+l) - Xj(i,k»)
'EB-A ai(Xj(i,n+l) - Xj(i,l»)
.EB-A
I
I
we get the desired result.
o 11.6.
DEFINITION: We say that a Banach space Y is K-finitely representable in a Ba-
nach space X if for every finite dimensional subspace E of Y there exists an isomorphism
~X with IIT1i·IIT-111 ~ K. Y is said to be crudely finitely representable in X if it is K-finitely representable in X for some K. It is said to be finitely representable in X if it is K
T: E
finitely representable in X for all K > 1. Using this definition one can state, for example, Dvoretzky's theorem as stating that £2 is finitely representable in any infinite dimensional normed space. DEFINITION: We say that a sequence {Yn}~=l in a Banach space Y is K-block finitely representable on a sequence {Xn}~=l in a Banach space X if for each N there exists N vectors {Zn}~=l in X of the form Zn = EiEO'Raixi, (]n disjoint, such that
for all choices of {an}~=l. We define now the notions of crudely block finite representability and block finite representability in an analogous way to that of the previous definition.
72
11.7.
COROLLARY:
Let
{Xi}~l
be any sequence satisfying the assumption of Theorem
11.4. Then there exists a sequence {Y;}~l (of nonzero vectors) in some Banach space Y which is block finitely representable on {Xi}~l and have the two properties. (i) It is invariant under spreading, i.e.
for all n, k l < ... < k n and all scalars
al,' ..
,an'
(ii) It satisfies
for all A ~ B, B finite, and all scalars {ai},EB. PROOF: Let Y be the completion of the space of all finite sequences endowed with the norm
where {Y;}~l is as in Corollary 11.5 (with en ....... 0) and let {Y;}~l be the unit vector basis in this space.
o 11.8. Note that (ii) of Corollary 11.7 implies
for all n, aI, ... ,an and all choices of signs. One can actually strengthen this to get the constant 1. THEOREM: Let {Xi}~l be a sequence as in Theorem 11.4. Then there exists a sequence {Zi}~l {x;}~l
of nonzero vectors in some Banach space Y which is block finitely representable on and has the two properties.
(a) It is 1 - spreading
for all n, k l < ... < k n and all scalars
(il) It is 1 - unconditional;
for all n and
al, ... ,
an'
all"
.,
an and
73 PROOF: Since block finite representability is a transitive property it is enough to prove the theorem for {Xi} = {yJ of the previous corollary. We may also assume IIYil1 = 1. We shall distinguish between two cases: Case 1:
Fix k and n. Let AI." ., An be intervals of integers each of length 2k,
(ji,ji +
Ai =
1, ... ,ji + 2k - 1) with
ji +
2k
< ji+1 for all i
= 1, ... ,n. Let
Then
Put, for i
= 1, . .. , n, Vi =
udluil .
Given a1,"" an and signs ~I."" ~n let Te be the isometry on span {Yi}~l given by
TeYj TeYj No~ice
= Yj = Yj+l
that for i such that
~i
= 1,
if if
j E Ai
and and
j E Ai
TeVi
= Vi
~i ~i
=1 = -1
and for i such that
So
If k is large enough with respect to n we get
I
Te(taiVi) - taivill
I
and in particular
Note also that
.=1
.=1
.
~ ~ IltaiVil1 .=1
~i
= -1
74
for all B
~
{1, ... , n} (since this holds for the y;'s).
The v,'s depend on n, denote v,
= vf.
For any sequence at, ... , am of scalars the sequence
(IIE~la,vfll)~=1 is bounded. We may pass to a subsequence M such that
exists for a dense set of all finite sequences of scalars
{a'}~1
(and thus for all the finite
sequences). The limit is necessarily a norm (note II{a;}~111 ~ maxla,\). The sequence {z;}~1 can be taken to be the unit vector basis in the completion of the space of finite sequences with respect to this norm.
Case 2: There exists an M such that IIEi::d/,1I :::; M for all n. (Note:
II Ei= dli II is a
nondecreasing sequence so that the two cases are inclusive). In this case we shall show that the unit vector basis of Co, the space of all sequences tending to zero, with the sup norm, is block finitely representable on {Y'}~I' Le., that for each e > 0 and k > 0 there are disjoint blocks ZI, • •• , Zk of {Y'}~1 such that
t
li ,=1
a,z,ll < (1 + e) max ja,J.lI z ,1I 1~,~k
(note that the property (i) of the {Y,} immediately implies IIE~=1 a,z,ll ;::: maxl$i~k la,I·lIz,II). First note that
Il
ta,Y,l1 :::; 2M m!lX la,l . 1~,~n
,=1
Indeed, let f be a norm one functional such that IIEi=la,y,1I = !(Ei=la,y,), then,
II~ a,y, I = ~ la,I!((sign a,)y;) n
:::; max la,l· !("(sign a,)y,) 1<,
:::; max la'!'11 L(sign a,)y,1I :::; 2M max la,l . 1~,~n ,=1 1~.~n The conclusion will now follow from the next lemma. LEMMA: Let (U,)i~1 be a sequence of nonzero vectors in a Banach space X with the property n'
La,u, ,=1
75 Then there exist n disjoint nonzero blocks Vi, ... ,V n such that
t
li i=l PROOF: For V
= I:aiui
~ VK· l:::;.:::;n max lai!·\Ivill
aiVil1
we denote \Ivlloo
= maxlail'
Iluili. If, for some 1 ~ k ~ nand
all al,'" ,an,
we are done. Otherwise there exist Vk E span{ U(k-l)n+ilf=l' k
= 1, ... , n, such that
Then for all al,' .. ,an,
II
t
akvk
I
~
K
I
t
akVk 1100
= K max laklllvklioo
l:::;k:::;n
~ Vi(. max lakl·llvkll . l:::;k:::;n
o 11.9. We conclude this chapter with two propositions which are finite dimensional analogues of results from this chapter. We shall use them in Chapter 13 below.
PROPOSITION: For every e: > 0 and positive integer k there exists an N that
= N(k,e:)
such
If n > Nand Yl, ... ,Yn is any sequence of norm one elements in some normed space then
+ e:)-spreading (i.e.
11I:}=lajYinjll ~ (1 + e:) 11I:}=1 ajYi mj II for any two subsequences nl < n2 < ... < nt, ml < m2 < ... < mt of 1, ... , k) or satisfies
there exists a subsequence Yi"." ,Yik which is either (1
k
(1 - e:)k! ~ Ave ±
L ±Yij
~ (1
+ e:)k!
.
j=l
PROOF: If this is not the case, let k and e: be such that for any n sufficiently large there exists a sequence
y~, ... y~
of norm one elements in some normed space for which the
conclusion of the proposition does not hold with these k and e:. For an appropriate subsequence of the positive integers, M, the limits
exist for all finite sequences al,"" at. Let {ei}~l be the natural basis in the space of all finite sequences and define
76 This is a seminorm. Taking the quotient by the null space (Le. identifying any two sequences for which the seminorm of their difference is zero) we get a normed space. As in the proof of Theorem 11.4 one can find a subsequence, M l , of the positive integers for which
exists for every sequence al," . ,ak. If for some al,' .. ,an with L:f=llai\ then clearly,
We then easily get that, for some
Sl
< ... <
Sk
i- 0
this limit is zero
in M l ,
and then that, if n E M is large enough, then
If
for all al,"" ak with L:f=llail
i-
0, we get similarly a subsequence of length k in some set
{Yf, ... ,Y;:} which is (1 + c)-spreading. This contradiction proves the proposition.
o 11.10. The last proposition is a finite dimensional analogue of Corollary 11.5. The proof is similar to the proof of Proposition 11.9 and we omit it. PROPOSITION: Let S, c > 0 and let n be a positive integer. there exists an N = N(n, S, c) such that If Xl, ... , XN are any N elements in the unit ball of a normed space
satisfying Ilxi sequence
Vj
Xj
= Xi.;
II
~
S for i
- Xi.;_1'
j
i- j. Then there exists a subsequence
= 1, ... ,n is 2 + c
unconditional.
Xi" .•. , Xi' n
such that the
12. KRIVINE'S THEOREM The main object of this chapter is to prove Krivine's Theorem (12.4) stating that the unit vector basis of some lp, 1
~ p
<
00
or Co is block finitely representable on any non
degenerate sequences in a Banach space. For later use in the next chapter we also need a version identifying the right p. This is given in 12.5.
12.1. We begin by recalling the notions of approximate eigenvalue and approximate eigenvector. Let X be a Banach space over the complex field and let T : X operator. A sequence {un}~=1
X be a bounded linear of norm one vectors in X is called an approximate eigenvector
with approximate eigenvalue A (E
(J;)
-+
if
The next lemma shows that any operator has an approximate eigenvalue. We sketch the proof briefly. LEMMA: Let T : X
-+
X be a bounded linear operator and let A be any member of the
boundary of the spectrum of T, A E Ba(T). Then A is an approximate eigenvalue.
An E (J; \a(T) be such that An -+ Athen II (T - AnI) -111 -+ 00. Otherwise AnI) -111·IA n- AI < ~ for n sufficiently large and it would follow that T - AI is invertible (with (T - AI)-1 = (T - AnI)-1E~=o(I - (T - AI)(T - An I)-1)n). PROOF: Let
II (T -
Let V n E X be such that
llvnll
= 1 and II(T - AnI)vnll
-+
O. Then also
II(T -
AI)vnll
-+
O.
o 12.2. We now prove a similar assertion for two commuting operators. PROPOSITION: Let X be a complex Banach space and let T, S : X
-+
Xbe two bounded
linear operators with T S = ST. Let A be any approximate eigenvalue of T. Then there exists J.I. E a(S) and a sequence {Un}~=1 of norm one vectors in X such that
PROOF: Let
LIM ai denote a Banach limit over bounded sequences, i.e.
i-oo
LIM is a i-oo
norm one, positive, linear functional over loo, the space of bounded sequences with the sup norm, with the two properties: (i)
(ii)
LIMi~ooai = LIMi~ooai+1
and
LIMi~ooai = limi~ooai for all convergent sequences {a;}~1'
Define a seminorm on (E EB X)oo by,
(See [D.S.], p.73).
78
Let N be the null space of this seminorm and let
Define T: X for
v=
(Vb V2,"
-+ .),
X by T(Xl,X2, ...) = (TX1,Tx2, ...) and define similarly S. Note that AV so that A is an eigenvalue of T. Let Y ~ X be the space of all
Tv =
eigenvectors of T with eigenvalue A. Since T S = S T, SY ~ Y. Let p, E 8u(Sly) and let Ui
= (Ui,b Ui,2"")
E Y be an approximate eigenvector for
li~ lim IISUi,n - P,ui,nll , n
p"
= limn Ilui,nll = 1 and
Le. lIuill
= lim IISui ,
- p,uill
=0
.
= lim IITui ,
- AUili
=0
.
Note that since Ui E Y also lim lim IITui,n - AUi,nll , n Finally, choose (i k ,nk)f=1 such that
and
and put
o 12.3. We turn now to the main part of this chapter.
THEOREM:
Let {Xi}~1 be an i-unconditional i-spreading sequence in some Banach
space X. Then the unit vector basis in lp, for some 1
< P<
00,
or in Co is block finitely
representable on {Xi}~I'
PROOF: For convenience of notation we change the index set from :IN to Q - the rationals in (0,1): Define a norm on the space of all sequences {ar}rEQ with only finitely many nonzero terms by
where rl < r2 < ... < r n contains all the indices r in Q for which a r =f. O. Denote this space by Yo. The fact that {Xi}~1 is I-spreading implies that the definition is consistent. Let Y be the completion of Yo and let {e r } rEQ be the unit vector basis in Y, i.e. er
= {8r
,8
}sEQ'
Since IIEf=laier; II = II 2:7=1 aixill for all rl < ... < r n , we have that, for all rl < ... < r n , < ... < Sn in Q and signs COb' .. , COn,
SI
79 and
We are going to show that the unit vector basis of lp for some 1 :::; p <
00
or of
Co
is block
finitely representable on {er}rEq and thus also on {Xi}~I. We extend Yo and Y to spaces over the complex field by defining
II ~rEq la r le rII
for {a r } ~
lI~rEqarerll
Consider the two operators on Yo defined by
=L
a rer /2
+
rEq
L are'~l
rEq
and =
L
are;
+L
rEq
are!..}!-
rEq
+ L are'~2 rEq
Note that
Ilxll :::; IITxl1 :::; 211 xll II xII :::; IISxl1 :::; 311xll , that
TS = ST and that T and S map nonnegative sequences to nonnegative sequences. By Proposition 12.2, given any approximate eigenvalue A of T there exist Jl- E u( S) and
{Yn};:"=1 E Y, IIYnl1 = 1, such that IITYn-AYnll n~O and IISYn-Jl-Ynll n~O. The positivity of T and S implies that IAI, 1Jl-1 and IYnl (= the sequence in Y attained from Yn by taking the absolute value of the elements of the sequence forming Yn) also satisfy the same property, so we may and shall assume that the original A, Jl- and Yn are nonnegative. Necessarily 1 :::; A :::; 2, 1 :::; Jl- :::; 3. We may and shall assume also that Yn E Yo. 12.3.1. IMPORTANT REMARK: The p in the conclusion of the Theorem will depend only
on the
Aabove;
actually 2 1 / p
= A (if A = 1 we
get co). This is important when we want to
identify the lp (or co) one gets (see 12.5 and the next chapter). We now continue with the proof. Let u, v E Yo satisfy max(supp u) < min(supp v) and let u~ and u~, ... , u~. be elements in Yo all with the same order distribution as Yn (i.e. if
Yn = ~~=lar,er, with Tl < ... < Te then uj j,n) and such that
=
~~=lar,e8' with SI
< ... <
max(supp u) < min(supp ui) < max(supp ui) < min(supp v)
for 0:::; i :::; 2 k
and max(supp ui) < min(supp Ui+l)
for 1:::;
i:::;
k
Se depending on
2 - 1.
80
Then
2k
Ilu + Lui + vll-Ilu + AkuO + vii n~O. i=1
AkYn11 n~O
This follows from the fact that IITkYn -
and the invariance of the norm under
spreading. We also have a similar statement with 3 k replacing 2 k , J1, replacing A and S replacing T. Let ui be as above and define for any finite sequence {a;}~1
111{a;}111 = li;.n
IlL
aiuill
where the lim is over an appropriate subsequence (one for which the lim exists for all finite {ai}~I)'
III . III is clearly a norm. Let Z be the completion of the space of finite sequences under this norm and let {Zi}~1 be the unit vector basis. Then {Z;}~1 is block finitely representable on {e r }rEQ' We are going to show that {z;}~1 is I-equivalent to the unit vector basis in either lp for some 1 ::; p < 00 or in co. First note that by (*) and its analogue for 3 k we get m+2 k
m
Laizi i=1
+
m
Laizi+ i=1
L
Lk
+
aiZi
m+~
i=m+1
+ Ak Zm+1 +
=
L aizi i=m+3 k +1
Lk
aizi
(**)
i=m+2 +1
m
00
Zi+
00
L aizi i=1
i=m+2 +1
i=m+1
L
m
00
Zi
00
Laizi+J1,kZm+1+ L aiZi i=1 i=m+2 k+1
for all {ail, m and k. It is also clear that {Zi} is I-spreading and I-unconditional. Equations
(**) and (* * *) together imply that m+2 k 3'
m
Laizi i=1
+
L
00
Zi
+
Lk
m
=
aiZi
i=m+2 3'+1
i=m+l
00
L
Laizi+AkJ1,lZm+1+ aiZi i=1 i=m+2 k3'+1
for all {ail, m, k and i. In particular, 2 k ,3'
L
Zi
= Ak . J1,l
.
i=1
CASE 1: An = J1,m for some n, m ~ 0 not both of which are zero. Assume, for example, 3m > 2 n . Then for k large enough 3 km > 2 kn +l. It follows that 3k~
2 kn + 1
Akn + 1 =
L i=1
Zi
::;
kn LZi = J1,km = A . i=1
81 Thus, A = 1 and it follows that {Zi}~l is isometrically equivalent to the unit vector basis of
co· CASE 2: 2k
2:
3l
*
An
#
JLm for all n, m
E IN.
Then both A > 1 and JL > 1.
Since
k
A 2: JLl we get that the function 2k
f(3 l
Ak ) = JLl '
defined for all k,l E IN, is increasing. Since both :~: ; and ll~~ ; are irrational (the irrationality of :~~ ; follows from An
# JL m for
{*
all n, m E IN), it follows that the set of their respective
integer multiplies mod 1 are dense in [0,1] and so g~ h,lElN and h,lElN are dense in IR+. This implies that the function f extends to a multiplicative function 1: R+ ~ R+. It follows that there exists a 1 :S p <
00
such that f(x) = x 1/ p ; in particular A = 2 1 / p and JL = 3 1/ p •
Next we are going to show that
for all k. We have already proved this for k = 2n ·3 m • For a general k, fix l E IN and let
n, m E IN be such that
2n
2n
-3 m < k < (1 + 3- l ) . - m 3 -
.
Then,
and
2n 3 l
2 n / P3 l / p
=
L
Zi
i=l
or 2
n p / •
3
l p /
:S 3
t;,~ II~ Zill :S 2
n p / •
3
l p /
+ 2n / p
•
t+~
Dividing by 3 - p - and using (* * * * *) we get
k 1 / p /(1
+ 3- l )1/ P :S
(2 n /3 m )1/ p :S
II~ Zill :S (2
n
/3 m )1/ p (1
+ 3- l / p ) :S k 1 / P (1 + 3- l )
.
Sending l to infinity we get
Finally, given M
= max mi
ai of the form ai
=
2 n ;/P3- m ;/p, i
II~ aiZil1 = 3- II~ 2 M p /
= 3-
M p /
II~ Zill = 3-
=
1, ... , l we get from (* * **) with
n ;/P3(M-m;J/
M p 1 / N /p
PZi
l
82
t
(
Laf
)
l/p
i=l
Since a;'s of this form are dense in the set of all coefficients the proof is complete.
o We conclude this chapter with three corollaries. The first one is Krivine's Theorem. 12.4. THEOREM: Let {Xi}~l be a sequence in a Banach space X with the property that for all n and all scalars al, such that for all i l
<
, an, not all of which are zero, there exist constants
°<
c
00
< in n
c::;
L ajXi;
::; C .
;=1 Then the unit vector basis in lp for some 1 ::; p
<
00,
or in
Co
is block finitely representable
on {Xi}~l'
PROOF: Apply Theorems 11.8 and 12.3.
o 12.5. The next proposition is a corollary to the proof of Theorem 12.3. This Proposition will be an important tool in the next chapter. PROPOSITION:
Let {Xi}~l be a normalized l-unconditional l-spreading sequence
some Banach space X with cotype q for some q
1 ::; p
<
for all s
00,
C
<
00
and e. \. 0, s ..:...
<
00.
In
Assume moreover that for some
00,
2: 1. Then the unit vector basis in lp is block finitely representable on
{Xi}~l'
PROOF: We begin as in the proof of Theorem 12.3, defining the spaces Yo and Y and the operators T and S. We are going to show that, under the assumptions of the Proposition,
>.
= 2 l / p is an approximate eigenvalue of T. Then continuing as in the proof of Theorem 12.3
(see Remark 12.3.1) we get the desired conclusion. To show that 2 1 / p is an approximate eigenvalue of T let 2 n _l
Yn
=
L
k=l
ak·2- ne k·2- n
83 where ak.2-"
= 2- l / p if k . 2- n = a· 2-(l+1),
a odd. That is, Yn takes the values
1 1 at 2
1 3 and 4 4
2- 1 / p at -
135
8' 8' 8 2-(n-1)/p at
and
7 8
1 3 2n - 1 2n ' 2n ' ... ' 2n
Now TYn takes the values
1 at 2- 1 / p
1 3 and 4 4 1 3 5 7 at and 8 8' 8' 8
2-(n-2)p at
1 3 2n - 1 2n ' 2 n ' ... ' 2n
and 2-(n-1)/p at
1 3 2n +1 ' 2 n +1 ,
••• ,
Consequently,
so that
Now, using the cotype q inequality, we get
Denote On = !P - _+_1_. Then On ~ 0 and p £2n-1
(if an ~ 1 then 1 + an and
+ a~ + ... + a~-l n~oo).
Finally, put Zn = Yn/IIYnll, then
Ilznll =
1
84
o The last proposition is a simple consequence of Proposition 12.5. We state it in the
12.6.
form to be used in the next chapter. PROPOSITION: Let On '" 0 and let {Bn}~=l be a sequence of finite dimensional normed spaces with a common estimate on their cotype q constant for some q < 00. Assume that B n
is spanned by a sequence Yi', ... ,y~ of norm one elements which is (1 + on)-spreading and C-unconditional for some C independent of n. If for every sequence t n -. 00, t n ::; n of integers
then
tp
> 0 and k there exists an > N there are k blocks of {Yi', ... ,y~} which are (1 + e:) -equivalent to the
is block finitely representable on {Bn}~=l' i.e., for every e:
N such that for n
unit vector basis of
t;.
THE MAUREY-PISIER THEOREM
13. 13.1.
Given an infinite dimensional Banach space X and 1 :::; P :::; 2 :::; q <
00,
recall that
Tp(X) (resp. Cq(X)) denotes the type p (resp. cotype q) constant of X (see 9.1).
Let Px qx
= sup{pj Tp(X) < oo} = inf{qjCq(X) < oo}.
Note, the sup and inf need not be attained so that X is not necessarily of type Px or of cotype qx. Most of this chapter is devoted to the proof of the following theorem of Maurey and
Pisier. This is a stronger version of Krivine's Theorem (12.4). 13.2. THEOREM: Let X be an infinite dimensional Banach space. Then lpx and lqx are
finitely representable in X.
We give a detailed proof of the cotype case (i.e. that lqx is finitely representable in X). A similar but somewhat more complicated proof can be given for the type case (see [M.S.]). We preferred, however, to sketch a different proof for this case which has the advantage of giving a reasonable estimation on the dimension of the embedded l~ space in the finite dimensional version of the theorem (see 13.12 below), but has the disadvantage of not working in the cotype case. We begin with the proof for the cotype case. For a fixed Banach space X we shall denote Tp(n) = Tp(X,n),Cq(n) = Cq(X,n), i.e., Tp(n) is the smallest T for which
13.3.
for all
Xl, ••• , X n
for all
Xl, • •• ,X n •
13.4.
LEMMA:
{Tp(n)}~=l
(a) (b)
E X. Similarly Cq(n) is the smallest C for which
For each 1 :::; P :::; 2 and 2 :::; q
<
00
the sequences {Cq(n)}~l and
are sub- multiplicative, i.e. for all integers n,k we have
86 PROOF: (a) Let {X;}i~1 be nk elements of X. For each 1 ::;
i::; nand fJ E [0,1], let us
define Xi(fJ) = L(i_l)k<j$ikTj(O)Xj. From the definition of Cq(n) we have
Integrating with respect to 0, we obtain
t!
IIXi(O)llqdO::; (cq(n W !
lit
Ti(t)Xi(O)lr dfJdt
nk
=
q
(cq(n))q! LTj(fJ)Xj
dO
]=1
which follows easily from the symmetry of the Rademacher variables. On the other hand, for each 1 ::; i::; n,
L II x jllq ::; (Cq(k))q (i-l)k<j$ik
!
q IIXi(O)ll dO
and the assertion follows immediately from the last two inequalities. (b) The proof is completely analogous to the proof of (a).
o 13.5. LEMMA: If N
>
1 is an integeT, and q ~ T ~ 2 is defined by Cr(N)
=N
1
/r-l/q, then
X is of cotype s for every s > q, i.e. qx ::; q. (Similarly, if N each s
> 1 and p::;
T ::;
2 is defined by Tr(N)
= N 1 /p-l/r,
then X is of type s, for
< p; we will not prove this part since, as we said before, our proof of Theorem 13.2 for
the type case will follow a different route). PROOF: From the monotonicity and submultiplicativity of the sequence {C r (n)}::='=I' it follows that there exists a constant C > 0 such that, for all integers n,
Now, let {x;}i=n be n elements of X, and with no loss of generality, assume that their norms
r
are in decreasing order. Then, for each 1 ::; k ::; n,
IIx.1I
~ j~~ I II ~ (~t I x;
~C .
~
x; II'
.-l(J
k-Jc,(k)
n
LTj(t)Xj j=1
(J
87
> q,
which follows from the triangle inequality in Lq(X). Thus, for each s
r
(t,II 'II·r ~ (t,.-: (J X
C
n
L rj(t)xj
·dt) I
j=1
L rj(t)xj ~ c· C,('....l (J j=1 n
8
)
dt
i
by Kahane's Theorem 9.2. So X is of cotype s.
o 13.6. COROLLARY: For each integer nand p,q with Px
(and Tp(n) 2: n.~
lot~·~n)
~
P ~ 2 ~ q ~ qx,
-i) .
13.7. LEMMA: If q ~ qx, then lim n _
oo
=~-
(Similarly, and again without proof, if p
> Px, then lim n _ -
q~' oo
lo~ ogTp(n) n
= .Pxl -
1). P
PROOF: It follows from the last corollary that lim log Cq(n)
n-oo On the other hand, let E:
> 0, r >
log n
>~_~ . - q
qx
qx, and let n be an integer. By the definition of Cq(n)
there exist Y1, ... ,Yn such that
(1- E:)Cq(n)
(I II~Yiri(t)lIq
dt)
~ ~ (~IIYillq)~ ~ n~-~ (~IIYillr)~
~ Crn~-~ (I II~Yiri(t)lr dt) ~ ~ C:n~-~ (I II~Yiri(t)llq dt) ~ for some constant
C:
(since X is of cotype r and by Kahane's inequality 9.2).
Hence,
Cq(n) ~ 1:~Crn1/q-1/r and, therefore, ---I'
1m n _
Since r
>
oo
10gCq(n) I < -1 ogn - q
1 - -. r
qx was arbitrary, the proof is complete.
o 13.8.
We now begin the proof of the cotype case of Theorem 13.2. If qx
Dvoretzky's Theorem 5.8, we are done. Assume 2
<
qx ~
00
= 2,
(the case qx =
00
then, by is much
88 simpler but we prefer to treat it in the general scheme), It suffices to prove the following claim. CLAIM: There exist constants 0 < " C < 00 and 0 < 0 < 1 such that for each E: > 0 and integer m there exist m elements Yl, ... ,Ym of X with
(i) 1 - 0 ::; IIYil1 ::; 1, i = 1, ... , m (ii) (Yl"'" Ym) is an unconditional basic sequence with constant::; C and is (1 + E:) -invariant to spreading. 1
(iii)
IIEi=lYjll ::;,' S'X
,
for all s ::; m.
Indeed, suppose that the claim is true. In the case qx =
00
the inequality (iii) implies
IIEl'ajYjll ::; , maxl$j$m lajl and span{Yj}.i=l is (C ·,)-isomorphic to l~. Use now the lemma in 11.8 to finish the proof in this case. If qx < 00, then we are in a position to apply Krivine's theorem. For each k, let E:k
= t,
say. Define
zj = Yj/IIYjll,
Yt, ... ,YZ J'
be k elements of X satisfying the claim with
= 1, ... ,k.
>
Then, for any qk
qx,
we shall have, for
every s ::; k,
•
1
sO« =
I>jrj(t) j=l
Thus, for any r, and each s ::; k,
L j=l Now it follows that if r
z]·k
<
'
- - S '-L_l X'.
-1-0
< qx then, for every sequence of integers
k
2:
Sk -+ 00,
=0.
On the other hand one can always find a sequence {qd~l such that qk > qx for all k, limk qk = qx and 1
k
II?
Cq• (X) =
1
Let r
> qx
then, if k is large enough,
1
sqx -ik
I·
1
00.
--.L_...!....
sJ!-' 2: s;,x ••
and
1
lims~' k
=
00.
89 Hence, lqx is indeed finitely representable in X, by Proposition 12.6. PROOF OF THE CLAIM: Assume first qx
13.9.
1
<
00.
Let k be given, and let us fix
1
2 ~ r = qx- ~ < q, where h > 0 is chosen so that k'-<x ~ 2. Put a = !(1- q:) = 2qXhlnk ' and choose n large enough, so that nO< 2: N(k,c) of Proposition 11.9. In the case qx = 00 we choose r so that k l/r ~ 2 and take a Xl,""
Xn
= 1/2.
Then, by the definition of Cr(n), there exist
E X such that
n ) I/r (n ) I/r (~ IIXill r > (1- c)Cr(n) JII ~ xiri(t)llrdr
13.9.1
.
With no loss of generality, assume that maxi IIXil1 = 1. Let 0 < 8 < 1 and put w = 1 - 8. Define, for each j 2: 1 and i 2: 1,
Xi
= x;jw j -
I
foriEAj'
,
If IAjl ~ nO<, we set Aj,o = Aj and do nothing else with this set. If IAjl
> nO<, then we choose
a subset Aj,I C A j for which
If IAj - Aj,d
> nO<, then there exists another subset Aj,2
with the same properties.
Aj
A;~d
C
=
Aj - Aj,I
Continuing in this manner we obtain a disjoint partition
= Aj,I U ... U Aj,mj U Aj,o,
where IAj,ol ~ nO< and for each 1 ~ p ~ mj, Aj,p C A;~2,
where A;~2 contains more than nO< elements, and for each T C A;~2,
(J
I iE~,p Xiri(t) Il
r
dt)
1
~
2:
IAj,pl-x
(Jlli~Xiri(t)llrdt)~ 1
ITI-x
We now split {I, ... , n} into two parts. The "bad" part is A'
= Uj Aj,o
and the "good" part,
its complement, is AU = Uj UI:9::>mj Aj,t. Let us show first that the "bad" part can be essentially ignored in inequality 13.9.1. Indeed, n
L i=l
Ilxill
r
=
L iEA'
Ilxill
r
+
L
Ilxill
r
iEA"
But a is chosen so that, by the property of Cr(n) in Corollary 13.6, nO< 1~ that
f
= o(Cr(n)r), and since
II L:~=I Xiri(t) Ilr dt (because maxllxill = 1, and the triangle inequality), we may assume
90 Thus, it follows from 13.9.1 that
and by the symmetry properties of the Rademacher variables,
(1 - e)'+lCr(n)'
1
r
L
XiTi(t)
dt
~
iEA"
II x ill r
L iEA"
so that we have a similar inequality to 13.9.1, involving only the "good" part of {I, ... , n}. Let us now introduce new symbols {BS}~=l for the subsets {Aj.t};.l~t~mj' Let us define
Us(lJ) = L XiTi(O), iEBs
8
~ l, 0 E [0,1].
Then
Integrating with respect to 0, and using the symmetry of the Rademacher variables, we obtain
Hence, there must exist at least one index i
~ 80 ~
l such that
dO -< (1 - e~r(lr r+ C r () n r "llxill L...J iEB so
r
.
Replacing the x;'s by the normalized x;'s, and taking e small enough, we have (observing that Cr(l) ~ Cr(n)): 1
r
L XiTi(O) iEB so But IBsol
~
d(O)) ,
~2(.L
!
Ilxill
.EB so
k, and by our choice of the number T, we obtain
(I .
L
.EBs o
XiTi(O)
r
),
~2IBsol~·
91 (in the case qx
= 00 we have formally
the same inequality because IB.ol~ S; k~ S; 2). Since
the set B. o is one of the sets Aj,t it is contained in some set A
== A}~~to
which consists of at
least N(k, 0:) elements, where N(k, 0:) is as in Proposition 11.9. By the choice of B. o ' we also obtain, for each TeA,
13.9.2
Now use Proposition 11.9. There exists a subset A
c A
of k elements which satisfies one
of the two alternatives there. However 13.9.2 eliminates the second one for k large enough. Therefore we obtain a set A of k elements that are (1 +0:) -invariant to spreading and, for each
TeA, satisfies 13.9.2. To complete the proof of the claim, it remains to show the existence of a set TeA, which is large enough and forms a "good" unconditional basic sequence. To do that we make use of Proposition 11.10. Let To
c
A be a subset such that {Xi};ETo is
contained in a ball of radius 8. Since they all have norm a functional x· EX' such that
Ilx'll
= 1 and
Ix'(x;)1
~
1 - 8, it follows that there exists
~ 1-
38 for each i E To. Then, from
13.9.2 we obtain
On the other hand for any scalars {ai} and any r ~ 2,
put M
=
g2Qx/(QX- 2 ).
(
4
~
) Ox -.
1 8 = 6) and To conclude, no more than a fixed number M of elements of {X;}iEA
Hence, (1- 38)jTol'1 S; 41Tolox1 and so ITol S;
1-36
.
Fix any 8
<
"31 (say
can be contained in a ball of radius ~. Hence, by a standard argument, there is a subset
il
TeA of at least points such that for any i # j in T, Ilxi - xjll ~ ~. By proposition 11.10, if k is large enough, we can find a subsequence Xi l , ... , Xi.~ such that the sequence
== Xi.; - Xi.;_l' j = 1, ... ,m, is unconditional with a constant S; 3, say, and all j S; m. By 13.9.2, we obtain, for each s S; m,
Vj
• LVj j=1
~3(1 Lj=1 vjrj(8)
r
1
~r 1
~3 (I Lj=1 xi.;rj(8) ~r +3(1 Lj=1 Xi.;_l rj(8) •
1
S; 24sox
IlVj II
r
•
r
1
~r
~ ~ for
92
and the claim follows, by setting
Yj
= ~Vj,
J:S m, (with 6 =
g,
"f
= 12, and C = 3). o
13.10.
We now turn over to the proof of the type case of Theorem 13.2. As we mentioned
above, rather than bringing a proof similar to the cotype case we prefer to briefly sketch a completely different approach due to Pisier [Pi2]. DEFINITION: For 1 < P :S 2, the stable type p constant of a Banach space X, denoted by STp(X), is the smallest constant C such that
E
11I:8ixill :S C (I: IlxilIP) ~
for all finite sequences {Xi} in X, where {8i}~1 is a sequence of independent symmetric pstable random variables normalized in L l . For p = 1 one defines the stable type 1 constant of Xin a similar way replacing EII~8ixill (which is
00
unless Xi
= o for all i) with (EII~8ixill!)2.
It is easily checked that STp(X) is also equal to the smallest constant D such that 13.10.1
E
lit
max Ilxill· n~ i=l 8iXili :S D· l:5.:5n
for all n and all Xl, ... , X n E X. Indeed if 13.10.1 holds and k l ,... , k n are any positive integers then for any Xl> ••• , X n , norm one elements in X, we have ((8 i ,j) is a double sequence of normalized symmetric p-stable random variables).
E
lit .=1
8i k ;
Xiii = E
t.=11=1t
8i,j Xi :S
n
:S D(I:k;)~.
i=l
Thus, if
Xl, ... , X n
are in X with norms
IIXil1 = (k;/s)~
It follows now that this inequality holds for any Xi'S,
for some integers k;'s then
Le.
STp(X):S D. The other side
inequality is trivial. 13.11.
Let {Ti}~l be a Rademacher sequence independent of {8i}~I. Then the sequence
{18iITi}~1
has the same distribution as the sequence {8i}~1 and by the triangle inequality
llllI:Ti(t)Xill
=
lIIIEI:Ti(t)!8ilxill
:S
II
E
III: Ti(t)1 8ilxi l = E III: 8iXili
93
We conclude that, for 1
< p S 2, Tp(X) S K· STp(X) for some universal K.
13.11.1.
On the other hand fix 1 S r < p S 2 and let {'Pi} be a sequence of independent symmetric r-stable random variables normalized in L l and independent of {r;} and {Oi}' Then for any
{Xi}
~
X
=
i IlL
=
Tp(X)EI L'PiOi IIXill1 = Tp(X)(L IOir Ilxill')~
l
E IIL'PiXill
E
l'Pilri(t)xill S Tp(X)E(L l'Pil P IlxillP)i
S Tp(X)(EI011') ~ (L Ilxill') ~ so that 13.11.2.
We conclude that, if Px < 2 then 13.11.3.
Px
13.12. THEOREM: For any 1
= in!
{1 < p < 2, STp(X)
= oo}.
< p < 2 and c > 0 there exists a 0 = o(c, p) such that If 1
1
p
q
(-+-=1) then X contains a (1
+ c)-isomorphic
copy of l;.
Theorem 13.12 implies the type case of Theorem 13.2. Indeed, if px = 2 we are done by Dvoretzky's Theorem. Otherwise STp(X)
= 00 for
any p > Px. So that X contains (1
+ c)-
isomorphic copies of l; for any k,c and p > Px. It follows that lpx is finitely representable inX. It is quite easy to show that STp(li') = n~ so that another corollary to Theorem 13.12 is Theorem 8.8. To prove Theorem 13.12 note first that we may assume STp(X)
<
00
(if STp(X)
= 00
X contains subspaces with arbitrarily large finite stable type p constant). Let Xl,"" Xn EX be such that maXl~i~nllxill S 1 and
Let Y be the random variable which takes each of the 2n (vector) values, Xll"" Xn,
-Xl," . ,-Xn with probability 2~ and let {Yi,j} i~l j~l be a double sequence of independent (X valued) randon variables all with the same distribution as Y. For a 0 to be chosen later, for k S o(STp(X))q and fot any a = (al"" ,ak) with E~=llaiIP = 1 we form k
00
Sa = LaiLj-iYi,j. i=l j=1
94
The proof of Theorem 13.12 is a standard consequence (see e.g. 4.1 and 8.8) of the following two propositions. 13.13. PROPOSITION: For an appropriately chosen 8 = 8(c,p),
(a) . for all a, b in the sphere of
l;.
(b) for all a in the sphere of
l;, where c
p
>
0 depends on p only.
13.14. PROPOSITION: For an appropriately chosen 8 = 8(c,p), and for any a in the sphere
of
l; 1 - P ((1 - c)EIISall ~
IISal1
+ c)EIISaID
~ (1
~ 2 exp( -8(STp (XW).
13.15. For the proofs of the two propositions we need a representation theorem for p-stable random variables from [L.W.Z.]. We refer the reader to [Mar.P.] for a proof. Let {A;}~1 be independent copies of an exponential random variable, Le.,
P(A; > t) = e- t , t > 0, i = 1,2, ... and let
;
r; = LA;. ;=1
PROPOSITION: For each 1 < p < 2 there exists a constant 00 > cp > 0 such that, if y is any symmetric random variable with ElylP = 1 and {y;} are independent copies of y which
are independent of the A; 's, then 00
1
X=L r7'y; ;=1
is a symmetric p-stable random variable with Let {r;,j} ;'::1' i
EIXI =
cpo
= 1,2, ..., be independent copies of {r;};'::1
of {Yi} ;'::1, ;~. Form 00
S;
=
1
LraY;,; , ;=1
k
Sa = La;S;. ;=1
which are also independent
95 It follows from the proposition above that f(Si) is a symmetric p-stable with Elf(Si)1 cp • (~Ei=1If(XiW) ~ and thus also f(Sa) is a symmetric p-stable with the same expectationcp • (~Ei=llf(Xi) JP) ~. The variable f(Ei=18iX;) is also a symmetric p-stable random variable.
Its expectation is Elf(E~=18ixi)1 = (Elxi!P)~. It follows that the two processes
have the same distribution (compare distributions of identical linear combinations of the two processes). In particular for each a in the sphere of t~.
13.15.1
To compare
- I~
EllSal1 with EIISa!l, one first shows EEf=l Iri~ r~
=
<
00
(we omit
this computation). Then
IEIISall- EliSa II I::;
t lailE~ Iri~ -r~
!IIYi,ill
k
::; i=1 L [ail· ~.
max
1~,~n
II xiii ::; k~
.~ .
Now,
So that, if k ::; S(STp(X))q for an appropriate S, we get that EIISall is approximately equal to
cpEl1 ~Ei=18ixill
and Proposition 13.13 is proved.
o To prove Proposition 13.14 we use Lemma 8.4. First notice that if S = of independent randon variables taking values in some normed space, and if generated by
Xl! ...
Ef=l Xi is a sum
1;
is the u-field
,Xi. Then
IIE(IISIII1;) - E(lISIII1;-dlloo ::; 211Xilloo, i = 2, ... ,to Indeed,
E(IISIII1;) - E(IISIII1;-d ::; E(II LXiii 11;) + E(ll xilll1;) i#i
E(II
L Xilll1;-d + E(II Xilll1;-d i#i
96 and a similar inequality holds for E(IISIII1;-d - E(II S III1;)· Applying Lemma 8.4 to
IISal1
and using the estimate above we get
EllSall1 > c)
P(IIISall -
:S 2 exp (
-8 . c
q
11{2. a,'. J'-1;P}.k Il q ,=1,J=1 P,OO
)
00
q
8 .c ) =2exp ( -~ and Proposition 13.14 follows now from Proposition 13.13.b.
o 13.16. One of the main problems in the local theory of normed spaces is to find the structure of the nicely complemented finite dimensional subspaces of a general normed space. In this connection one may wonder whether one can choose the copies of the e;x's and
e~x's
in X
to be uniformly complemented. The examples of L oo and L 1 show that this is not the case. Moreover, a recent example of Pisier [Pis3] shows the existence of a Banach space X which admits uniformly complemented
e; - s for no 1 :S p :S
00.
We conclude this chapter with a
proposition showing that under some additional assumptions one gets a positive result. We refer the reader to Chapter 9 for the definition of the projections Rad n and recall that if sUPnllRadnl1 < 00 on X then for each 2 :S q :S 00 1
1 q
(-+-=1). p
In particular ...!... Px
+ ...!... qX
= 1.
PROPOSITION: Let X be an infinite dimensional Banach space. Assume
sUPnllRadnl1 < 00
on X and assume in addition that X· is of type Px'
definition of Px' is attained). Then, for some K <
00
and for each ~
k, X contains a K - complemented subspace which is (1
(i.e.,
the sup in the
> 0 and positive integer
+ ~)-isomorphic
to
e;x'
PROOF: Let p = Px, q = qx. By Theorem 13.2, X contains for each n and ~ > 0 a sequence e1, ... , en with
It follows from the Hahn-Banach theorem that one can find
and xi(ej)
= 8;j,
1:S i, J':S n. In particular for i
xi, ... ,x~
in X· with
implies now that for each k, if n is large enough, there exists a subsequence that the sequence
II x: II
:S 1
=f i, 1 :S IIxi - xiII :S 2. Proposition 11.10
xi,,· .. ,Xi,.
such
97 is 3-unconditional. Let d j = ei,j j = I, ... ,k, then yi(d j ) = Oij 1 projection P from X onto spun{dj }j=l by
:s;
i, j
:s;
n. Define a
k
Px
= Lyj(x)dj . j=l
Then,
IIPxIl
~ (1+ e) (t,IY;(X)I') \ k
=
(1 + c)sup{1
L
Ujyj (x) I;
L
IUjlP = I}
j=l 1
:s; 3(1 + c)llx ll sup{l
k
L Tj(t)Ujyj
dtj
o j=l
L \UjIP = I}
k
:s; 3(1 + c)llxIITp(X*) sup{(L IUjIPllyjIlP)l/Pj
L
IUjlP = I}
j=l
:s; 6(1 + c)Tp(X*)llxll and
IIPII :s; 6(1 + c)Tp(X*) .
o
14. THE RADEMACHER PROJECTION In this chapter we return to the subject of estimating the norm of Rad n projection onto the subspace {E~1Ti(t)Xi;
-
the natural
E X} of L 2 (X). The main theorem here (14.5),
Xi
due to Pisier, states that sUPn
seen to hold also in the real case).
14.1. Given two functions f E L 2({-1, 1}n, X) and g E L 2({-1, 1}n,
L
* g(c) = Tn
f(c' 6)g(6)
c E {-1, 1}n
6E{ -1,1}n
where c·6 is the coordinate by coordinate multiplication. Note that if f g
= L~{1, ... ,n} WAG:A, where {WA}A~{1,... ,n}
L
f *g =
= LA~{1, ... ,n} WAXA,
are the Walsh functions, then WAG:AXA·
A~{1, ... ,n}
Note that the norm of the operator
f
f * g is smaller than or equal to IIgIIL•.
--+
For t :::: 0 let n
gt(c)
= II (1 + e- t . ci),
c
= (cl, ... ,cn) E {-1, 1}n
i=1
and let for
St/=f*gt
fEL2({-1,1}n,X).
Then {Sth~o is a semi-group of norm one operators on L 2 (X) (check). For j = 1, ... , n, let Ej be the conditional expectation operator onto the u-field generated by the functions independent of the j-th component, i.e.,
where the 1 and -1 stand in the j-th place. It is easily checked that n
St =
II (Ei + e-t(I -
Ei)) .
i=1
One can write
n
St
=L k=O
e-ktWk .
99 Note that WI
= Ei=1 (I -
E;)ITj#;Ej is no other than Rad n .
The main step in the proof that if t 1 is not finitely representable in X then sUPnllRadn!l < 00,
consists of showing: that, in that case, St extends to an analytic semigroup {SehEV in
some sector
V={zE
(J;;
Rez~O,
largzl
with sUPeEv!lSe11 ~ K, for some 0 < ill, K < 00 independent of n. Then it is quite easy to complete the proof by showing that WI (the coefficient of e-e in the expansion of Se to power series in e-e) is bounded by a bound independent of n. We begin with two lemmas. 14.2.
LEMMA: If
t1
is not finitely representable in X then there exists an integer N
and a real number 0 < P < 2 such that for all N -tuples PI, ... , PN of commuting norm one projections on X one has
PROOF: If this is not the case then for all Nand C > 0 one can find N norm one commuting projections PI, . .. ,PN and a norm one x E X such that
We are going to show that in that case x, PI x, ... , PN:z: is nicely equivalent to the unit vector basis of +l .
tf
First notice that for all
C 1 , ••• , C N
= ±1
Indeed, let
A and for C
~
= {kick = -I},
B
= {kjck = I}
{l, ... ,N} put Pc
=
II Pk
(P0
= I) .
kEC
Then, N
II (I k=1
II (I - Pk) II (I - Pk) kEA = L (-l)ICIPc II (I - Pk)
P k) =
kEB
C<;B
kEA
100 and N
(-l)IBIPB
II (I-Pk) = II(I-Pk)- L(-l) CPC II(I-Pk).
kEA
k=l
kEA
C~B
C¢B
Consequently,
IIPB
II (I -
Pk)xll ~ 2 N - e - (2 1B1 - 1)2 1A1
= 21A1 -
e
kEA and
II
II (I + Pk) II (I -
kEB
kEA
Pk)xll ~ IIPB
II (I + Pk) II (I -
kEB
~ 2 1B1 1IPB
Pk)xll
kEA
II (I -
Pk)xll ~ 21B1 (2 1A1 - e)
kEA N ~ 2 (1 - e) which proves (*).
The vector rrk'=l (I + ekPk)X is equal to x + Ek'=l ekPkX plus 2N - N - 1 vectors each of norm at most one. It follows that, for all e1,'" ,eN = ±1, N
Ilx + L
ekPkxll ~ 2 N (1 -
e) -
2N + N
+1=
N
+ 1 - e. 2 N .
k=l To evaluate IIEk'=oakPkxll from below for general real coefficients, put k
ek
= sign
ak,
= O, ... ,N, then
II~ akPkxl1 = II~ eklaklPkxl1 ~ ~ laklll~ CkPkXII-II~ ek(~ lati-lakl)Pkxll ~ (N + 1- e' 2 ~ lakl- ~ (~Iatl-Iakl) = (1 N
)
e' 2N ) ~ lakl
that is, if e . 2 N < 1, x, P 1 x, ... , PNX is 1-;.2 N -equivalent to the unit vector basis of lfH over the reals. This means that ll, over the reals, is finitely representable in X. Since this implies that complex II is finitely representable in X(check), the proof is complete.
o 14.3. Lemma 14.2 implies in particular that if II is not finitely representable in X then there
exists a p < 2 and an M < projections Pl, ... , Pn ,
00
such that for all n and all n-tuples of commuting norm one
101
Given such an n-tuple of commuting norm one projections Pi>"" Pn , we define a semigroup {Sth~o of operators by
n n
St
=
+ e-t(I -
(Pj
t ~0 .
Pj)),
j=l
LEMMA: Assume £1 is not finitely representable in X. Then there are p < 2 and M <
00
such that, given any family PI, ... ,Pn of commuting norm one projections and forming the semigroup {Sth~o as above, there exists a norm 1·1 on X satisfying (a)
IStXI::; Ixl
(b)
1(/ - St}xl ::; plxl for all t
for all t ~ 0 and x E X
(e) Ilxll::; Ixl ::; Mllxlj
~ 0
and x E X and
for all x EX.
PROOF: The operator St is a convex combination of projections from the set
c ={
n Pk,
A
kEA
Indeed,
n(e- t / +
~ {I, ... , n}}
.
n
St =
(1- e-t)Pj)
j=l n
=L
e-(n-k)t(l - e- t )
k=O and
L
n Pj
IAI=kjEA
n
L e-(n-k)t(l_ e-t)(~) = 1 . k=O
It follows that II St II
::; 1.
Convexity considerations imply also that
n l
(I - Sti) ::; M . /
j=l
for all t1, . .. , tl
~
O. One way to see this is to introduce, for 1 ::; j ::; £, 1 ::; k ::; n, independent
random variables, €j,k(W), with
P(€j,k(W) P(€j,k(W) Then,
= 1) = 0)
n
IIj(w)
=
II (Pk + €j,k(W)(/ k=l
Pk))
=
II {k'{i,k=O}
Pk
102 and
Stj
=/
II; (w)dP(w) ,
= 1, ... ,e.
j
We get that, for each w, l
II(l-II;(w)) :5.M./
;=1 (note, for each w, II; (w), j = 1, ... , n is a family of commuting' norm one projections so that the remark before the statement of the lemma applies to it). Consequently, l
II / (1 -
II; (w))dP(w)
;=1
l
=
II (1 -
/
II;(w))dP(w)
;=1 l
:5./ II(I-II;(w)) dP(W):5.M./.
;=1 We introduce a new norm on X by
e,
where the supremum is taken over all m, t 1 , •.• , tl and properties follow immediately from the two inequalities
IIStl1 :5. 1
and
II II (I -
r1,""
Stj) II :5. M.
r m • The three required
pl.
;=1
o 14.4.
The main tool in the proof of Theorem 14.5 below is an analytic extension theorem,
due to Beurling and Kato, for strongly continuous semigroups of operators. We recall that a semigroup {Sth>o of operators on X is said to be strongly continuous if StX
----'0 x for every tx EX. If V is some open cone in
THEOREM: For all 1 :5. p < 2 there exist constants tP > 0 and K < 00 such that if is a strongly continuous semigroup of contractions on some complex Banach space
{Sth~o
satisfying sUPt~0111 where
Stll :5. V
p
< 2. Then
= {z E
OJ
{Sth~o admits an analytic extension
Re z
~ 0,
larg
and {SehEV satisfies sup
eEV
IISell :5. K
.
zl < tP}
{SehEV
103 We postpone the proof to Appendix IV. 14.5. THEOREM. Let X be a (complex) Banach space. Then £1 is not finitely representable in X if and only if sUPnI!RadnIIL.(x)
PROOF:
< 00.
By Lemma 9.10 (see also 9.11), sUPn,mIIRadnIIL.(t;n) =
possible to show that IIRadnIIL.(t~) ~ £1 is not finitely representable in X.
..;rogn.)
00.
(It is actually
It follows that if sUPnIIRadnIIL,(X) <
00
then
If £1 is not finitely representable in L 2(X)then, defining, as in the beginning of this
chapter n
St =
IT (E; + e-t(I -
E;)) ,
;=1
we get by Lemmas 14.2, 14.3 and Theorem 14.4 that {Sdt~o admits an analytic extension
{Sehev to some sector
v=
{z E
zl < ¢>}
with sup
eev
!lSdl::; K
where ¢> > 0 and K are independent of n. Necessarily,
n
Se
= IT (Pj + e-e(I -
n
Ej»
j=l
= L::e-keWk k=O
(since the right hand side defines an analytic semigroup which coincides with St for t E
t
~
m,
0). Recall that WI = Rad n . For a
= 11" /tg¢>, a + ib E V
for all
-11" ::;
b ::;
11".
Also, it is easily checked that
It follows that
It remains to check that if £1 is not finitely representable in X then it is not finitely representable in L 2 ({-1,1}n,X) uniformly in n, Le., that it is not finitely representable in
L 2 ([0, 1], X). This follows from the fact (Theorem 13.2) that if £1 is not finitely representable in X thim X has some type p > 1. Then, by 9.12, L 2 ([0, 1], X) has the same type and consequently £1 is not finitely representable in L 2 ([0, 1], X).
o 14.6. We now state a theorem estimating IIRadnll for a general finite dimensional space.
104 THEOREM: Let X be a finite dimensional normed space 01 dimension k. Then, lor all
n,m
IIRadnIIL.({-l,l}~.X)~ (e + 1)log(d(X,l~)
+ 1)
.
In particular,
lor some absolute constant K. PROOF: Note that Radnl = 1* 9 where 9 = E~l rio For any g, III * gIlL.(X) ~ II/IIL.(X) . IIgIIL If X is a Hilbert space and 9 = III * gilL. (X) ~ 1I/IIL.(x) . maxlcAI· It follows that, in any space X 1 •
ECAWA we also have
For -1 ~ 0: ~ 1 let g,,(c) = TIi=l(1 + O:ci) then g" ~ 0, f g" = 1, g" = Ek'=oo:kEIAI=kWA. For J.L E (C[-I, 1])* let 9 = f~l g"dJ.L(O:). Then
III * gIIL.(x) That is, the operator Tgi = we get that if J.L satisfies
~ i l l III * g"IIL.(x)dJ.L(O:) ~ 1I/IIL.(x)IIJ.L11 .
1* 9 has norm ~ IIJ.LII· Since 9 = Ek=o(f~l o:kdJ.L(a))EIAI=kWA
then
Bernstein's inequality (see e.g. [Zy]) states that for a degree l polynomial p on [-1,1]
Ip'(O)1 ~ l max Ip(t)l. -l~t~l
It follows from the Hahn-Banach theorem that there exists a J.L E (C[-l,l])* with
IIJ.LII
and with f~l tdJ.L(t) = 1 and f~l tkdJ.L(t) = 0 for k = 0,2,3, ... ,l. Let a> 1 and define
I/(A) = aJ.L(aA). Then
III/II
~ a· e,
For k > l we have
1 1
-1
tdl/(t)
=1
and
1 1
-1
tkdl/(t)
=0
for
k
= 0,2, ... ,i .
~
l
105 so that
and we get
IIRadnll = II T (Ei= 1 r;) II S a.e+a-l.e.d(X,e~). Taking a = e and e = [log d(X,e~)l
+ 1 we get the desired result. o
14.7. We conclude the chapter with a corollary to Theorem 14.5 which shows that the
eI,eoo
example in 9.11 is essentially the only example to the case where duality between type of a space and cotype of its dual does not hold. COROLLARY: Let X be a Banach space with px > 1. Then p:x 1 PROOF: If Px > 1 then
sUPnIIRadnIIL,(X) <
00
e1 is not finitely
+ q:X~ = 1.
representable in X so that by Theorem 14.5
and the corollary follows from Lemma 9.10.
o
15. PROJECTIONS ON RANDOM EUCLIDEAN SUBSPACES OF FINITE DIMENSIONAL NORMED SPACES In this chapter we combine results from Chapters 4 and 14 to prove the existence of nicely
complemented almost euclidean subspaces in a large class of normed spaces (Theorem 15.10 below). After some introductory results, we bring in 15.4 the general scheme of the proof.
15.1. We start by introducing two norms on the space B(i~. X) of operators from normed space X. In both cases we shall omit the subscript n from the notation.
15.1.1. The i-norm: For A : i~
--+
i~
into a
X define
where IL is the normalized Haar measure on that one can also write
sn-l -
the unit sphere of i~. Recall (cf. 9.5.1)
n
i(A)
= (Ell L 9;Ae;11 2 ) 1/2 ;=1
where (e;)i'=1 is the unit :vector basis of i~ and (9;)~1 are independent, symmetric, gaussian random variables normalized in £2.
15.1.2. The r-norms: Fix an orthonormal basis u
= (Ul •... , un)
in i~ and define
where (r;)i'=1 are the Rademacher functions. EXERCISE: There is an obvious one to one correspondence between the set of all orthonormal bases and O(n) - the orthogonal group. Show
where IL is the normalized Haar measure on O(n). The norms i and r u are not ideal norms in the usual sense of this notion (see [Pie]), however they share some of the properties of ideal norms. In particular
= IIAII
(i)
i(A)
for any rank 1 operator (use the second definition of i(A)). And,
(ii)
i(A B G) :::; IIAII ·i(B) ·IIGII for any normed spaces X, Y and operators A: X --+ Y, B : i~ --+ X, G: i~ --+ i~ (first check it for unitary G then use the fact that any norm one operator on i~ is a convex combination of unitary operators).
107 We remark in passing that an ideal norm is a norm, defined on the space B(V, X) of operators from V to X for any two finite dimensional normed spaces V and X, which satisfy (i) and (ii) for any diagram
15.1.3.
The dual norm:
Given any two finite dimensional normed spaces X and Y and a
norm i on B(X, Y), one defines a norm i* on B(Y, X) by i*(B)
= sup{trace
BA;
A E B(X, Y),
i(A):S; I} .
Clearly, (B(Y, X), i*) is the dual space to (B(X, Y), i) with duality < A, B >= trace BA. THEOREM: Let X and Y be n-dimensional normed spaces. Let a be a norm on B(X, Y) and let a* be the dual norm. Then there exists an operator W: X ~ Y such that a(W) . a*(W-l) = n.
15.2.
REMARKS: 1)
Note that for any W, n = trace W· W-l :s; a(W) . a*(W-l).
2)
One can prove F. John's Theorem 3.3 as a corol1ary to Theorem 15.2, see [Lew] and
[Pel· PROOF OF THEOREM 15.2: Let W be the operator in B(X, Y) for which the maximum max{ldet Ulj is attained (the set U: X ~ Y, a(U) is clearly invertible). Let U: X
~
U: X ~ Y,
= 1 is compact).
a{U) = I} We shal1 show that a*(W- 1)
=n
(W
Y be arbitrary and consider {3
which is wel1 defined for smal1 {3
= Idet(a(W + tU)-l . (W + tU)) I ItI·
By the choice of W, {3 :s; Idet WI. On the other hand
= a(W + tU)-n ·Idet WI·ldet(I + tW-1U)1
.
Hence Idet(I + tW-1U)11/n :s; a(W
+ tU) :s;
a(W)
+ ta(U) = 1 + ta(U)
.
Using the Jordan canonical form it is easily seen that 1
-tr n
(W-1U) _ I' det(I + tW-1U)1/n -1 - 1m < a (U) . t-O t -
o
108 r~
15.3. The next le=a relates the LEMMA: Let T: X Then
-+ l~
norm of T to the r u norm of its adjoint.
and let
U
=
(Ul,"" un) be any orthonormal basis in l~.
PROOF: By the orthogonality of the Rademacher functions, for any S: n
l~ -+
X
n
< TSUi,Ui >= L
trace TS = L i=l 1
< SUi,T*Ui >
i=l
1
< tr,(t)Sui' tri(t)T*Ui > o i=l ,=1 :S ru(S)ru(T*) ,
=
and we conclude that
To prove the other inequality recall the duality between L 2 (X) and L 2 (X*). It follows that for some {Y;}i=l' {y..d
with
IAI'"
A~{l •...• ft}
n
Ly,ri(t) i=l
ru(T*)
=
lit
+
L
Define S:
l~ -+
X by SUi
= Yi.
L.(X)
=
ri(t)TOUill
.=1
L.(X)
n
< T*Ui, Yi >
.=1
n
< ui,TSUi >= L
i=l
ru(S) =
t
Then
ru(S)r~(T) 2:: trace TS = L and
:S 1 ,
YAWA(t)
IAl;o!l
lit •=1
< T*Ui,Yi >= ru(T*)
i=l
:S IIRadnIIL.(x) .
ri(t)Yill L.(X)
o 15.4. As a corollary to Theorem 15.2 and Le=a 15.3 we get that there exists aT: with r u (T)r u (T-l*) :S IIRad n ll' n. The result of Appendix II implies that for any q > 2 and
Xl, • •• ,X n
EX,
l~ -+
X
109
where K depends on q and Cq(X) - the cotype q constant of X - only. A similar inequality holds for X*. We conclude that 15.4.1. where K depends only on q, q*, Cq(X) and Cq• (X*). Moreover, using Exercise 15.1.2, one can easily get 15.4.1 with a universal constant K. We now introduce an euclidean norm on X by taking its unit ball to be the image by T of the unit ball of l~. Using the method of Chapter 4 and in particular Proposition 4.6, we would like to find a large dimensional almost euclidean subspace of X which is nicely complemented. The quantities M r and ~l(T) are closely related (the first is the median and the second in the quadratic mean of IITxl1 on sn-l). Instead of relating everything to M r and using Chapter 4 we prefer to work with l(T) and imitate the method of Chapter 4. The production of the complemented euclidean spaces will be done in three stages. Let
=n
X C Y, dim X show that T- 1 the
r: norm.
:
X
and let T: -t
l2
-t
X be as above. In the first step (Lemma 15.5) we
l~ can be extended to S: Y
-t
l~ in such a manner as to preserve
Then one gets the same estimate for l(T)l(S*) as in 15.4.1. The second step
(Corollary 15.8) is to pass to a large dimensional subspace E of
l~
on which we have good
estimates for II TIE II and IISi~1I in terms of the l norms of T and S* respectively. The last step (Le=a 15.9) is an imitation of Proposition 4.6 which shows that one can produce a good projection onto T E. 15.5.
LEMMA: Let a be a norm defined on B(X, Y) for some finite dimensional normed
space X and all finite dimensional normed spaces Y, which is injective in the sense that if Y c Z, with i the inclusion map, then for all T: X - t Y, a(T) = a(iT). Then for any Y C Z and any R: Y - t X, R admits an extension R: Z - t X with a*(R) = a*(R). Note that r u and l are injective. PROOF OF LEMMA 15.5: The injectivity of a implies that (B(X, Y), a) is naturally a subspace of (B(X,Z),a). Consider the functional f on (B(X,Y),a) defined by f(T) =
trace RTwhose norm is a*(R). By the Hahn-Banach theorem one can extend f to a functional Ion (B(X, Z), a) with the same norm. Any such functional is given by an operator R: Z - t X acting by i(T) = trace (RT). It is easy to check that R extends R. Finally, a* (R) = 11111 = 11111 = a*(R).
o The second step (the passage to a subspace on which we have good estimates for both
IITIFII
and IISI~II) will be done in two propositions. The proof of the first one is deterministic and of the second one probabilistic. We recall that fh(X, k) is the gaussian cotype 2 constant of X on k vectors (see 9.4). Clearly (see 9.12), .B2(L 2(X),k) = .B2(X,k) so one gets easily that .B2((B(l~,X),l),k) =
.B2(X,k) .
110
15.6. PROPOSITION: Let U: i~ -+ X be any operator. Then for every 1 ::; k ::; n there exists an E ~ i~ with dim E ~ n - k and
PROOF: First notice that if {Pi};=l are pairwise orthogonal projections in i~ then 15.6.1.
min i(UPi)::;
l~i~k
~.B2(X,k)i(U).
yk
Indeed, k
1
k
\~}~ki(UPi)2::; ~i(UPi)2::; .B2((B(i~,X),i),k)2li(~ri(t)UPi)2dt ::; .B2(X,k)2i(U)
llll~ri(t)PiI12dt::; .B2(X,k)2i(U) .
Let now {P;}~l be a maximal set of pairwise orthogonal projections with rank Pi such that, for all 1 ::; i ::; m,
By 15.6.1, m < k. Let F projection P with ImP
= (ImE~lPi).L. ~
=1
Then, by the maximality of {Pi}~l' any rank one
F satisfies
Finally,
11U1F11 = sup{11U Pili P:
i~ -+ F, P a rank one orthogonal projection} .
For rank one operators the operator and the i-norm coincide, hence,
and F is the desired subspace.
o We apply the proposition for both T and S' (see 15.4. S is the extension of T-l to Y in a manner which preserves the that dim F l , dim F 2 ~ ~n and
r: norm) with k = in to obtain two subspaces F
l ,
F 2 such
111 The subspace F = F 1 n F 2 has dim F ~ ~n and
PROPOSITION: Let U: l2' -> X be any operator. Let Jl, be the normalized Haar measure on sm-l. Then for any C ~ 1
15.7.
Jl,{x E S
2
m-l
IlUxll >
j
PROOF: The function x
->
3C { C l(U)2} y'ffll(Un:s 2 exp - 11U11 2
IIUxll is a Lipschitz function with constant :S
11U11.
Let M
denote the median of this function. Then
So ~l(U) ~ M and by Chapter 2,
o We apply this proposition to F = l2', m Jl,{x E Fj
~ 1-
IITxl1 :S
2 exp
~ ~
and to the operators TI F and SIF to get
3 y'ffll(TIF) and
(4(32(;
IIS'xll :S
n)2 ) - 2 exp
3 y'ffll(SIFn
(4(32(;:,
n)2 )
The method of Chapter 2 now implies 15.8. COROLLARY: with T and S as in 15.4, there exists a subspace E
for some absolute constant c
~
l2 with
> 0, on which
This concludes the second step. The third step consists of a standard lemma resembling Proposition 4.6.
112
LEMMA: Let X a 1 - 1 operator, and let
15.9.
~
Y be Banach spaces with dim X = n <
s:
Y
->
£2 be an extension of T-1
Then there exists a projection P from Y onto TF with
IIPII
:<: :;
00.
Let T: £2
->
X be
Let F S;;; £2 be any subspace.
•
IITIFII
IISi~.II.
PROOF: Let Q be the orthogonal projection from £2 onto F and let P = TQS. clearly a projection onto T F. Now, for any x E Y,
IIQSxl1 2 =
Pis
(QSx, QSx) = (Sx, QSx)
IIxll IIS·QSxll IlxIlIISI~IIIIQSxll ,
=< x, S·QSx >:<:::; :<: :; so that
IIQSxl1
:<: :;
IIxll
.
IISI~II
Finally,
IIPxl1 = IITQSxl1
:<: :;
IITIFII·IIQSxl! :<: :;
IITIFIIIISI~lllIxll
. o
Combining the consideration in 15.4, Lemma 15.5, Corollary 15.8, Lemma 15.9 and Theorem 14.5, we get THEOREM: Let Y be an infinite dimensional Banach space with type p > 1 (i.e.
15.10. Tp(Y) <
00
for some p > 1).
Then there exists a constant C <
and positive integer k, an integer N
= N(e,k)
such that if X
contains a k-dimensional subspace Z with d(Z, £~) :<: :; 1
+e
00
and, for each e > 0
Y and dim X 2: N then X
~
and there exists a projection of
norm :<: :; C from Y onto Z.
PROOF: The space Y must have some finite cotype otherwise it contains, by Theorem 13.2, uniform copies of £;:"'s and thus does not have any type> 1. Lemma 9.10 implies that
for some q <
00
and C <
00,
for all finite dimensional y
Theorem 14.5, R = sUPNIlRadNI!L.(Y) < Let X
~
l
~
Y, Cq(Y/O) :<: :; C. Also, by
00.
Y be an N dimensional subspace and let y l be any finite dimensional space with
X ~ y ~ Y. By Theorem 15.2 we get an operator T: l
e:
->
X with ru(T)r~(T-l) = N
1
is any orthonormal basis). Let S be an extension of T- to an operator S: y
with r~(S)
= r~(T-l)
l
--+
£~
(which exists by Lemma 15.5). Then, by Lemma 15.3, ru(T)ru(S·) :<: :; ru(T)IIRadNllr:(S)
:<: :; Rru(T)r~(T-l)
= R· N
.
By the relation between the Rademacher and Gaussian averages (Appendix II) 15.10.1.
(u
113 where K depends on the cotypes and the cotype constants of Y' and Y'·. Corollary 15.8 now implies the existence of a subspace E of if with
on which
IITIEII This implies (since IISI~II ~
II(TITE)-lll)
IISi~;\I ~ 16· K .
that
d(E,i~) ~ 16· K and, by Lemma 15.9, that TE is 16· K-complemented in Y'. Holder's inequality implies easily that
for any normed space Z and any q> 2. Thus, since Cq(Y'),Cq(Y") ~ C for some q and C independent of Y', n
-> 00 as N -> 00. An application of Theorem 4.3 produces a subspace Z of E of dimension k, where k -> 00 as n -> 00, which is (1 + e)-isomorphic to i~. Of course this subspace is 16· K-complemented in E and thus (16· K)2-complemented in Y'. Finally, a compactness argument, let us pass from Y' to Y.
o
APPENDIX I ISOPERIMETRIC INEQUALITIES IN RIEMANNIAN MANIFOLDS by M. GROMOV
1. GENERALITIES ON RIEMANNIAN MANIFOLDS 1.1. Local Riemannian structures. A Riemannian C r -structure 9 on an open subset U in the Euclidean space lR n is, by definition, a Cr-map u -+ gil., U E U, which assigns to each point U E U a positive definite quadratic form (i.e. an Euclidean metric) gil. on lR n . If Xl, ... ,Xn is a fixed basis in lR n , then 9 is determined by the n( n2+I) scalar products gu.(Xi,Xj ) which are Cr-functions on U. One defines next the 9 -length of each Cl-map
where uEU
I'
stands for the derivative
*:
I:
[0,1]-+ U by
[0,11 -+ lR n. For example, if gil. is constant in
= lR n , then the family gil. reduces to a single Euclidean structure on lRn
and the above
notion of the length agrees with the Euclidean one. Finally, for connected domains U, one defines the distance distg(UO,Ul) between points Uo and Ul in U as the infimum of the lengths of Cl-curves between Uo and Ul, which are C l _
I: [0,1] -+ U such that f(O) == Uo and f(l) = Ul. This dist g is called the Riemannian metric associated to 9 and it obviously satisfies the ordinary metric axioms. Moreover, the
maps
topology associated to dist g agrees with the induced Euclidean topology in U. Observe that dist g1
=
distg.implies gl
= g2·
Indeed, (gu.(X,X))~
= lime_og-ldistg(u,u + gX)
for all
U E U and X E lR n • However, only very special metrics on U are of the form dist g for some g. For example, no metric dist(UbU2) = Ilul -
u211
for a non-Euclidean norm
II
lion lR n is
ever Riemannian. EXAMPLES. (a) Let U be an open interval, say (a,b) in lR l
.
Then each Riemannian
structure is given by a single function gil. and the metric space (U, gil.) is isometric to the
115
ordinary (Le. with the metric lUI - u21) interval (a', b') in mI for a' = b' = f:(g .. )!du, where one may take an arbitrary fixed point in (a,b) for c.
f: (g.. )! du and
(b) Take m2 with the standard Euclidean structure go and let U be the unit disk in m~. Put g.. = 4h(u)go for the function h(u) = (1 - dist;o(U,0))-2 = (1 - go(u,u))-2. This is called the Poincare metric and the metric space (U, dist g ) is called the hyperbolic plane. Every map f: [0,1) -+ U for which go(f(t),f(t)) -+ 1 for t -+ 1 clearly has infinite length. This implies (by an easy argument) the completeness of the hyperbolic plane. The remarkable (though easy to prove) property of this 9 is the invariance under conformal transformations of the disk U. In fact the isometry group of (U,dist g ) is (easily seen to be) locally isomorphic to the multiplicative group of unimodular 2 x 2 matrices. Riemannian manifolds. A metric space (V, dist) is called an n-dimensional Riemannian Cr-mani/old if it is locally isometric to some (U,dist g ). Namely, for each point v E V there are some neighbourhoods U' and U" c U' of v in V and a homeomorphism
1.2.
H of U' on an open subset U dist g (H(UI),H(U2))
c mn
= dist(UI,U2)
with some Riemannian Cr-metric dist g such that for all points UI and U2 in U".
EXAMPLES. (a) Let V be a Cr+l-smooth n-dimensional submanifold in some Euclidean space mq , q :2: n, with the standard Euclidean metric. Recall that a subset V c mq is a Cr+l-submanifold if there exists, for each Vo E V, a linear n-dimensional subspace mn ~ L c mq , such that the orthogonal projection, say P: V -+ L, homeomorphically maps a small neighbourhood U' C V of Vo onto an open subset U eLand such that the inverse map p-I: U -+ U' c mq is a Cr+l_map of U into mq. Define dist(V1JV2), for Vl,V2 E V, by taking the infimum of the lengths (measured by the Euclidean metric in mq :::> V) of C I _ curves in V between VI and V2' Then the resulting metric in V is Riemannian. Indeed the orthogonal projection mq -+ L r:::: mn sends the tangent n-planes T,,(V) c mq of V at points v E U'(which are close to vo) onto L. This brings the Euclidean structure on T,,(V) down to L, say to g.. on L for u = P(v), and the projection P: U' -+ U clearly is an isometry near Vo for the resulting Riemannian metric on U. This metric on V c mq is called the induced Riemannian metric. The corresponding Riemannian distance in V is typically greater than the distance in the ambient space m, since straight intervals in mq with the ends in V not always lie in V. REMARKS. (a) A deep theorem of Nash claims that every Riemannian manifold V is isometric to some manifold in mq (where q must be rather large compared to n = dim V). For instance, the hyperbolic plane can be realized by a C1-submanifold in m3 , but no such C 2 -realization exists. However, every bounded ball (or rather disk) in the hyperbolic plane is isometric to a Coo-submanifold in m3 • It is unknown if there is a global realization of the hyperbolic plane in m4 , but this is possible in m6 (see [Gr 3] for an account of the theory of isometric imbeddings of Riemannian manifolds). Unfortunately, the existence of isometric imbeddings V -+ mq has no application to the actual study of the Riemannian geometry of
116
V. (a') The above example obviously generalizes to smooth submanifolds V in an arbitrary Riemannian manifold W. Namely, the restriction of the length functional on curves in W to those in V induces a Riemannian structure in V from W. (b) Take the unit sphere sn-l C IR n with the induced Riemannian metric (from the Euclidean structure in IR n ) and let pn-l denote the projective space which is obtained from sn-l by identifying the pairs of points sand -s in sn-l. Since the involution s ....... -s is an isometry on sn-l, one obviously has a unique Riemannian metric on pn-l for which the quotient map sn-l ....... pn-l is locally an isometry. Observe that pn-l (unlike sn-l) admits no isometric C 2 -realization in IR n • (In fact, the space pn-l for n ~ 3 admits no isometric C 2 -imbedding to IR n + 2 , but p2, for instance, can be isometrically Coo-embedded into IR 5 ). (b') Take linearly independent vectors X1, ... ,Xn in IR n and let £ ~ 7J,n be the free Abelian group (lattice) generated by these vectors. Then the quotient space, called a flat torus, y n = IR n / £ inherits a Riemannian metric from the Euclidean one in IR n for which the quotient map IR n ....... yn is locally an isometry. The space of (isometry classes of) fiat tori is (for a natural topology) a fairly complicated locally compact space of dimension n( n2+l). No fiat torus admits an isometric imbedding into IR m for m ~ 2n - 1 but many are embeddible into IR 2n (e.g. 2-tori go into IR 4 and split n-tori which are isometric products of circles obviously go into IR 2n for all n). (c) One usually defines a Riemannian structure on a smooth manifold Vasa family (or a field) of Euclidean structures in the tangent spaces Tv(V) which is an assignment to each v E V of a positive quadratic form gv on Tv(V). For example, if V is a smooth submanifold in IRq, then each tangent space Tv (V) is realized by the n-plane which is the n-dimensional affine subspace in IRq (geometrically) tangent to V at v. Each Tv(V) can be brought to the origin in IRq by the parallel translation Tv(V) ....... Tv(V) - v. By restricting the Euclidean form in IRq to this (now linear) subspace Tv (V) - v in IRq one gets a particular field of forms in Tv(V) which gives the induced Riemannian structure in V described above in the language of length. (c') Let Xi = Xi(V) , i = 1, ... , n, be linearly independent tangent vector fields on an n-dimensional manifold V. Then one may uniquely define a metricg on V by putting
g(Xi,Xj ) = Oij, where Oii == 1 and Oij == 0 for i #- j. This is, in fact, a very general procedure. For example, if V is homeomorphic to IR n then every Riemannian metric on V comes this way with some fields Xi. Indeed, the space IR n with any Riemannian metric admits, by elementary topology, a frame of n orthonormal vector fields Xi. Another (by far more interesting) example arises from Lie groups. Namely, if V is endowed with a structure of a Lie group, then each tangent vector at the neutral element e E V uniquely extends by the left group translations to a left invariant field on V. Thus each basis of vectors
Xi C T.(V) extends to a frame on all of V and we obtain with the above what is called a left
117 invariant Riemannian metric on V. In other words, left translations extend each Euclidean structure in T.(V) to a Riemannian metric on V for which left translations are isometries. These metrics are easy to define, but not at all easy to understand. For example, if V is the multiplicative group of non-singular (m x m)-matrices, then T.(V) equals the linear space
M rn = IR n , n = m 2 of all (m x m)"matrices. The standard Euclidean structure in this IRn uniquely extends to a Riemannian metric in the space V of non-singular matrices. We suggest to the reader to evaluate the Riemannian distance between two matrices with given entries. Equidistant translates of hypersurfaces in m n • Consider a C2-smooth hypersurface Vo (i.e. a submanifold of dimension n - 1) in IRn with the Euclidean metric and let V t , t E IR, be a smooth one-parametric deformation of Vo, that is a C2- map, say F: Vo x IR ---> IR n, such that the map F on Vo = Vo x 0 equals the original embedding
1.3.
Vo "-> IR n . The deformation is called normal to Vo if the derivative ~(v,O) E IR n is normal to the tangent hyperplane Tv (V) c IR n for all v E V. We want to study the deformation of the induced metric, now called gt on the manifold Vo = Vo x t which is mapped into IR n by v ---> F(v,t). (The image F(Vo x t) may not be a submanifold in IR n , but this is rather irrelevant for our present purpose). The length length g, (C) of each curve C C Vo by definition of
gt is the Euclidean length of the curve c ---> F(c, t). This is equivalent to saying that gt(X, X) for each vector X E Tv(Vo ) equals the squared Euclidean length of the image of X under the differential of the ~ap F on Vo = Vo x t c Vo x IR. Here each Tv(Vo) C IRq is viewed as a vector space where the point v E Tv(Vo) is taken for the origin. PROPOSITION-DEFINITION (Gauss).
There exists a unique continuous map which
assigns to each v E Vo and each vector v = Vv normal to Vo at v a symmetric linear operator A v : Tv(Vo) ---> Tv (Vo) with the following property. If F is an arbitrary deformation for which ~(v,O) = v, then g~=o(X,Y) = 2go(A v X,Y), where X and Yare arbitrary fixed tangent vectors in Tv(Vo) and g~ stands for the derivative Furthermore, A"v = aA v for all a E JR.
'?!to
PROOF: Identify a small neighbourhood of a fixed point in Vo with a domain U C IRn - 1 , let Ul,"" Un-l be the Euclidean coordinates in U and let Oi = be the corresponding derivations (fields) in U. Then the scalar products gt(Oi, OJ) by the definition of the induced
-k:
metric are where < ,
> is the Euclidean scalar product in
(using ~(v,t)
=v
for t
IR n. Since
< V,OiF >= 0 for t = 0, we have
= 0 and operating with OJ) d < dtojF,oiF >= - < v,oiojF > at t = 0
and so
g~=O(Oi,Oj) =
:t < OiF,OjF >
=-2 .
118 Thus the operator A ... is uniquely defined by go(A ...a"aj) = - < v,a,ajF > at t = O.
o 1
REMARK. Fix a point Vo E Yo, let Vo be a unit normal to Vo at Vo and let IR C IR be the I-dimensional subspace parallel to vo. Take the orthogonal hyperplane IR R - 1 ~ L C IR R which is parallel to T vo (Vo) C IR R and observe that the orthogonal projection P: Vo -+ L R
diffeomorphically sends a small neighbourhood Uo C Vo of Vo into a domain U C L. Denote by PI the orthogonal projection IR R -+ IR 1 and observe that Uo C IR R equals the graph of the function
f
= PI p- l
:
U
-+
IR I. Furthermore,
< vo, a,aj F >= a,a;! ,
u, in this U and, hence, - ~g~o equals the second differential
for the Euclidean coordinates 2
d f of fat
Uo
= P(vo)
E U.
Next consider a hypersurface V o C IR R which is an inside tangent (relative to vo) to Vo at vo. That is V o contains vo, the tangent hyperplane Tvo(Vo ) C IR R equals Tvo(Vo) and the corresponding function 1 satisfies 1 ~ f near uo. Then d 21 ~ d 2 f and we conclude to the following (simple but useful) inequality
See the first chapter in [Mil for additional information. Next we turn to the following normal geodesic deformation F: Vo x IR -+ IR R which isometrically sends each line R = v x IR c Vo x IR onto the straight line in IR R normal to
Vo at v. Such a map exists if and only if Vo is normally orientable in IR R (unlike the Mobius band in IR3 ) which is always the case for small neighbourhoods of points in Yo. We want to study the second derivative for normal geodesic maps F. We denote by A(t) the operators
g;'
A ... assigned to the hypersurface F(Vo x t). Again, our consideration is local near a fixed point v E Vo and we only allow those values t for which the map F diffeomorphically sends U' x t for a small U' around v onto a smooth hypersurface in IR R. This is the case (by the implicit function theorem) if and only if the differential of F is injective on the tangent space Tv,t(Vo , t). The following theorem shows this property to fail exactly for those t which equal the reciprocals of the eigenvalues of the operator A(O) at v and which are called the principal curvatures of Vo at v relative to the unit normal v THEOREM (Gauss-Weil).
= ~(v,O).
The derivative 01 A (t) in t satisfies A'(t) = -A 2 (t) ,
where A 2 is the ordinary square 01 the operator A. PROOF. Since <
dJ: ,ajF >= 0, we have
d
d
Id
dt
t
Z t
I,
< -a,F,ajF >=< a,F, -d ajF >= --d < a,F,ajF >= -zgt(a"aj) = gt(A(t)a"aj) .
119 Hence,
2
d
d
d2
d
< dt o.F, dt ojF >= dt 2 < o.F, ojF >= 2 dtgt(A(t)o., OJ)
= 2g~(A(t)o.,Oj) + 2gt (A'(t)o., OJ)
= 4g t (A 2 (t)o.,Oj)
+ 2gt (A'(t)o., OJ)
,
which equals, by the above, 2g t (A 2 (t)o.,Oj).
o EXAMPLES. (a) Let Vo be the unit sphere sn-l C
m. n and let F(s, t) = s(l- t) be the
geodesic deformation corresponding to the interior normals. Then, clearly, g~ = -2(1 - t)go and therefore A(t) = -(1- t)-l Id. Thus all principal curvatures of the sphere of radius 1- t equal -(1 - t)-l for the interior normal direction. They become infinite at t F collapses sn-l to a single point.
= 1 as the map
m. n which bounds a compact region c m. • Consider the interior normal geodesic deformation F: Vo x m.+ -+ m. n and let (b) Let Vo be an arbitrary closed C 2 -hypersurface in
V+
n
r(v) E R+ = m.+ x v for v E Vo be the first singular point on m.+ x v where the map F fails to be a local diffeomorphism. (We know by the above that -(r(v))-l equals the smallest
= {v,tlv E Vo,t < r(v)} C Vo x m.. of the difference V +\F(W+) c mn equals
eigenvalue of A(v,O)). Consider the open subset W+ PROPOSITION. The n-dimensional measure zero.
PROOF. Take a point x E V+ and let r., be the radius of the greatest ball B., in V+ around x. The boundary oB., is an inside tangent to Vo at some point v = v(x) E Vo. Hence,
x = F(v,r.,) and r(v) ~ r.,. The set F{v,r(v)} c almost all points x E V+ are contained in F(W+).
m.n
obviously has measure zero, and so
o EXERCISE. Let Vo be a closed convex hypersurface whose principle curvatures relative to the exterior normal are :S 1. Show the normal geodesic map F(v,t), where ~ is the
exterior normal field, to be injective on Vo x (-1,00) c Vo x m.. Show that the convex region bounded by Vo contains a unit ball. Next consider a convex subset V+ with smooth convex
boundary Vo = OV+ which has the principal curvatures> 1. Define Vt C V+ as the set of those v E V+ for which dist(v, Vo) ~ t. Show that each set Vt is convex. Prove, moreover, that Vt is the intersection of convex subsets with smooth boundaries whose principal curvatures > (1 - t)-l. In particular, the minimal non-empty Vt is a single point, say, v in V+. Show that V+ is contained in some ball of radius 1 in m. n . 1.4. Normal deformation of the Riemannian volume. Each Riemannian manifold of dimension n carries a canonical measure (volume) which is uniquely defined by the following two axioms:
120 (1) (2)
The volume of the unit cube in IR n equals 1. IT I: VI -+ V2 is a distance decreasing map of VI onto V2 , then Vol V2 where dim VI
:::;
Vol Vb
= dim V 2 = n.
This is obvious since each continuous Riemannian structure gu in U C IR n can be approximated near each point Uo E U by the Euclidean structure guo (which is constant guo in
U). The following computational formula for this volume is equally obvious. Let
11-0 be the
Euclidean Haar measure in IR n and let l1-u be the Haar measure associated to the form gu on IRn (for which the measure of the gu-unit cube is one). Then the ratio r(u)
= l1-u/l1-o
is
a continuous function in u and Vol U = I r(u)dl1-o. Moreover, one defines the Jacobian of a smooth map f: VI -+ V2 between Riemannian manifolds, say J(v), v E Vb as the absolute value of the determinant of the differential D,: T,,(VI) -+ T",(V2 ) for
Vi
= f(v),
relative to
the Euclidean structures g" in T,,(VI) and g~, in T",(V2 ), where g" and g~, are the Riemannian structures in VI and V2 respectively (which by definition are Euclidean structures in the tangent spaces). IT f is a bijective map, then Vol V 2
= Iv,
J(v)dv where dv is the Riemannian
measure in VI. EXERCISES. (a) Show every compact manifold V to have finite total volume Vol V. (b) Show the hyperbolic plane
H2
to have Vol
H2
= 00.
(c) Let go be a Euclidean structure in IR 2 and let a continuous function h(u) on IR2 equal (lIulilogllull)-2 outside a compact subset in IR2 for lIuli = distgo(u,O). Show that the Riemannian metric 9 = hgo on IR2 is complete and Vol(R 2 ,g) < 00. (d) Show every non-compact Lie group with a left invariant metric to be complete of infinite volume. (e) Let SLnIR be the group of unimodular (n x n)-matrices with a left invariant metric and let S L n 1l be the subgroup of the matrices with integer entries. Show the quotient manifold SLnIR/SLn'lL to be complete of finite volume (and noncompact for n ~ 2).
REMARK. The double coset space SOn\SLnIR/ SLn'lL is naturally homeomorphic to the space of flat tori yn of unit volume. Now, with the Gauss formula gHX, Y) = 2go(A v X, Y) for normal deformations we immediately see the following formula for the derivative in t of the volume Volt = V oI g, (Uo) for all domains Uo C Vo V ol~=o =
r trace Avdvo
Juo
for the volume element dvo of the metric go (which is the usual volume of a submanifold in n
IR n ). The trace of A v relative to a unit normal is called the mean curvature of Vo C IR and it clearly equals the sum of principal curvatures. For example, the mean curvature of the unit sphere sn-l C IR n equals n - 1 for the exterior normal and it is -(n - 1) for the interior normal.
121
°
In order to apply this formula to t iwe express the volume element of gt on n F(Vo x t) C IR by dVt = J(vo, t)dvo for the Jacobian of the map F. Then for all t E IR,
Vol~ = ~ {
dtl uo
J(vo,t)dvo
= {
(trace Av(vo,t))J(vo,t)dvo . Juo
This is equivalent to the relation dJ(vo, t) dt = J(vo,t)Trace Av(vo,t) which holds on the greatest interval a(vo) ~ t ~ b(vo) around zero where the Jacobian does not vanish. Then one exresses the above by d -log J(vo,t) = Trace Av(vo,t) , dt
and the Gauss-Weil formula implies d2 l i d -d2log J(vo,t) = -trace A~(vo,t) ~ ---(trace A v (vo,t))2 = ---(-log J(vo,t))2 . t n-l n-l dt
Now, a straightforward computation shows that JVo, () t
~
1
+(
t Trace Av(Vo,O))n-1
EXERCISE. Show that J(vo,t) = Det(1
n-l
+ tAv(vo,O)).
1.4.A. THEOREM. (Paul Levy [Lev]) Let Vo be a closed hypersurface in IR n whose mean curvature relative to the interior normal everywhere is ~ li for some number li < 0. then the
region V+ bounded by Vo has V ol V+ ~ - nnl/ Vol Vo . (In fact the equality holds if and only of radius -Jil( n - 1).
If Vo is a round sphere PROOF.
Vo x IR+ -+ IR n be the interior normal geodesic map and let
Let F:
°
W+ = {va E Vo, ~ t < r(vo)} C Vo x IR+ be the maximal open subset on which F is locally diffeomorphic. (This is equivalent to J(vo,t) > for (vo,t) E W+). Then J(vo,t) ~ (1 + ~l)n-l which implies r(vo) ~ f = - nil for all (vo,t) E W+. Hence, Vol F(W+) ~
°
J1
r("o)
Vo
0
Since meas(V+\F(W+))
J(Vo,t)dvodt ~ Vol Vo
1" 0
(1 +
lit n- 1 __ )n-l = --=-Vol Vo . n - 1
nlJ
= 0, the proof follows. o
1.5.
Normal deformations in Riemannian manifolds.
Let V be a COO-smooth Rie-
mannian manifold of dimension n which is complete as a metric space. A Cl-map f: IR -+ V is called a geodesic if dist(f(to),f(t)) = Ito - tl for all to E IR and for all t E IR close to to. Let Vo be a C 2 -smooth normally oriented hypersurface in V. Then there exists a unique
122 (with the given normal orientation) normal geodesic map F: Va x IR
-+
V for which the
curve .l'I"oxIR: R = va x IR -+ V is a geodesic normal to Va at va for all va E Va - (see [Mil. [G.K.M.], [C.E.]). We define as above the family gt of the induced Riemannian metrics on Va = Va x t and we study g~ and g; at t = 0 as earlier. First we define the operators A v = A(va, t). where the normal direction v in V is understood as the image of the field on Va x IR under the differential of the map F, by setting gt(A v , X, Y)
=
ft
~gaX, Y) for all
pairs of tangent vectors X and Y in Va. PROPOSITION-DEFINITION (Riemann).
There exists a smooth map v
-+
K v which
for each va E Va C V assigns to the unit vector v E T"o(Vo), normal to T"o(Vo) C T"o(V) for all va E V, an operator K v : T"o(Vo) -+ T"o(Vo), called normal curvature operators, such that ftA v = -A~ - K v at t = 0 for all normal vectors. Furthermore, the operator K v depends onlll on v (and on the Riemannian metric in V, of course) but not on Va. That is, if Va and Ve: have a common tangent space at some point va, that is va E Va n Ve: and T = T"o(Vo) = T"o(Ve:) C T"o(V), then the operator K v : T -+ T defined bll K v = -(ftAv+A~) for vonve:, automaticalillsatisfll the same relation for Ve:. (Observe that the operator A v , unlike K v , does depend on Ve:.) The proof (by a straightforward computation) can be found in any textbook on Riemannian geometry (e.g. [G.K.M.], [C.E.], where an equivalent language of Jacoby fields is employed). EXAMPLES. (a) Let V be the round sphere sft in IR ft +1 of radius R and let Va be a round sphere Sft-l in Sft whose points are all within (geodesic in Sft) distance ro from a fixed point v E V = sft. then, clearly, the metric gt on Va = Va x t satisfies gt = P sin 2 [(t + ro) / R]go for p = sin 2(ro/R). Then g~ = (i[sint)[Ocos!1P-)gO and so A v = (R-l ctg t)[O)Id. Next, A~ = (_R-2 - R-2ctg(tij?)2)Id and so K v = R- 2Id. Thus the sphere Sft ofradius R has (constant) curvature R- 2 • (b) The exponential map. There exists a unique map T,,(V) -+ V for each point v E V, called exp: T" (V) -+ V, which isometrically sends each straight line 1.. in (T" (V), g,,) ~ IR ft through the origin onto a geodesic 'Y in V through v which is tangent at v to t. (See [Mi], [G.K.M.], [C.E.].) It follows that each ball Be C T,,(V) around the origin of small radius e > 0 is diffeomorphically sent onto the e-ball Be C V around v. Hence, the boundary Se of Be for small e is a COO-smooth hypersurface in V, whose interior normal geodesic map
F: Se x IR+ -+ V diffeomorphically sends Se ~ [O,e) onto Be\{v}, while the map F(s,e) collapses Se to v. This implies, like in the Euclidean case, the following property of the normal geodesic map F of an arbitrary hypersurface Va in V.
If for a fixed point Xo E V the function dist(xo, v), v E Vo, assumes a local minimum at some point va E Va then F(vo, r) = Xo for r = dist(xo, va) (and for the obvious choice of the normal Vo at vo) and the map F isiocalill diffeomorphic at (vo,t) for 0 ~ t < r. COROLLARY.
Let Va be a closed hllpersurface in V which bounds a compact re-
123 gion V+ let W +
C
V, let F:
= {vo E Yo,
0 ~ t
Vo x m+ -+ V be the interior normal geodesic map and C Vo x m+ be the maximal open subset on which F is locally
< r( t)}
diffeomorphic. Then meas,,(V+ \F(W+)) =
o.
REMARK. We shall need below a simple generalization of this fact to compact subsets
V+ C V with possibly non-smooth boundary. Namely, let Vo be the topological boundary of V+, let V& C Vo be the maximal open subset which is a COO-smooth hypersurface and let E = Vo\ V& be the complementary singular part of Yo. The singularity E is called negligible for V+ if no compact subset V~ C V+ with COO-smooth (n -I)-dimensional boundary ever meets
E. EXAMPLE.
Take a domain D in S2 bounded by a smooth curve C in S2 and let
V+ C m. :) S2 be the cone over D from the origin. The boundary Vo = av+ is singular at the origin unless C is the great circle and this singularity is negligible if and only if D contains 3
no hemisphere. Now, for Vo with a negligible singularity we define the map F outside E, that is
F: V& x
m.+
-+
V and the image of the corresponding W~ C V& x
m.+
obviously covers
almost all of V+ . EXERCISES. (a) Show that the curvature of the hyperbolic plane everywhere is -Id. (b) A hypersurface Vo is called convex (concave) for a given normal orientation if the operators A... are positive (negative). Show that normal geodesic deformations preserve convexity if the (symmetric) operator K ... is negative for all
Similarly, these deformations
11.
preserve the concavity if K ... ::::: O. RICCI CURVATURE.
Define Ricci
11
for all unit vectors
11
in T(V) byRicci
Trace K .... For example, the round n-dimensional sphere of radius R has Ricci
11
11
=
= const =
(n - I)R- 2 • The following (by now obvious but important) formula generalizes the volume
deformation property from
m."
to all V.
l.S.A. LEMMA. The Jacobian of the map F satisfies
d dt'og J(vo,t) = Trace A ... (vo,t) ,
and d22'og J(vo,t) = -trace
&
A~(vo,t) -
Ricci
11
~
Observe that the equality here holds for spheres Vo
__ I_(dd
log J(vo,t))2 - Ricci
n-l
t
= S,,-1
C V
11.
= Sri.
I.S.B. THEOREM. (Paul Levy). Let a closed submanifold Vo with a negligible singularity in V bounds a compact region V+ C V such that the mean curvature (that is Trace .4... for the interior normal) of the non-singular locus V& ~ Vo everywhere is ~ Ji, for a given Ji E m (of any sign now) and let the Ricci curvature of V everywhere::::: (n -1)R- 2 for some R > O. Let be the ball of the (geodesic) radius ro in the round sphere Sri C m,,+1 of radius R, where
V;
124
= -ll/(n - 1). (The boundary Vo* V';)Vol Va T'h en, V 0 I V+ $ (VolVol V. .
0$ ro $ 1rR such that R-1ctg!j
mean curvature
= -) p, •
01
this ball has constant
a
PROOF. The Jacobian J(vo,t) of the interior normal map on w~
= {va E vri,
0 $ t < r(t)} C
vri x IR+
is majorized by the Jacobian of the corresponding map for Yo*, that is J(vo,t) $ J*(va,t), where the Jacobian J* (obviously) depends only on t but not on va' This majorization is obtained by a direct computation with the above formulae. It follows that r(vo) $ ro for all Vo E Vb and that
Vol V+ $ Vol F(W~) $
ii
i ira .()
V 01 Vb $ V 0I 0 v.'V o · o
v~
r(vo)
J(vo,t)dvodt
0
* _ Vol Vb Vol t dvodt V 0 I v.'
J
v;
0
This is obvious for those who have confidence in their Riemannian geometry. A novice is invited to check all the details step by step by comparing with the Euclidean case. EXERCISES.
(a)
Let V+ be a compact region in V with possibly non-smooth
boundary. Say that the mean curvature of the boundary is ~ II if V+ is the intersection of a decreasing sequence of regions with smooth boundaries which have mean cur-
ll. Take the e:-neighbourhood of the boundary, say Ue(aV) C V and define Vol av = limin!e.....oe:-1Vol(Ue(aV) n V+), where Vol on the right-hand- side denotes the n-dimensional Riemannian volume (or rather the measure) which is obviously defined for all Borel subsets in V. Generalize the above theorem to these regions V+.
vatures
~
(b) Let V have Ricci 1/ ~ 0 for all unit vectors and B 2 of radii R 1 and R 2 ~ R 1 have
1/
E T(V). Show that concentric balls B 1
(n - 1)R- 1, for R > 0, and let Bi and B 2 be the balls of radii R 1 and R 2 in the sphere SR of radius R. Show that (b') Let Ricci
1/
~
Vol B
Vol Bi
1 -< --Vol B - Vol'B 2
2
and prove as a corollary that
Vol V $ Vol
SR
and
diam V $ diam
SR = 1rR .
(c) Generalize the above (*) to the case Ricci 1/ ~ (n - 1)R- 2 for R < 0, by replacing the sphere SR by a pertinent hyperbolic space (or, alternatively, by writing explicit formulae in place of Vol Bi/Vol B 2).
125 (d) Show all balls in a complete simply connected manifold of negative curvature (that is, the operators K v are negative) to be convex (i.e. to have the convex boundary for the exterior normal). (e)
Let a complete manifold V have K II ~ R-2Id for some R > O. Show that the
complement to each ball in V of radius ~ 1rR is a convex subset in V(i.e. it is an intersection of regions with smooth convex boundaries).
2. ISOPERIMETRIC INEQUALITIES 2.1. Inequalities for Banach spaces. The classical isoperimetric inequality in rn. n claims that among all domains V+ c rn. n of a given volume the round ball B n has the minimal
(n - I)-dimensional volume of the boundary, (1) where C n = Vol Bn/(Vol sn-l)n/n-l for the unit sphere sn-l = aBn. This can be equally expressed with the characteristic function !+ of V+ (which is 1 on V+ and 0 outside) by Sobolev's inequality which relates the Lp-norm of functions on rn. n for p = n~1 to the L 1norm of the differentials df,
r (Jrn. rn. n
n
Ifl n / n -
1
) (n-l)/n :S c~n-l)/n Jr
Rn
lidflldu
(2)
for all functions
f on
measure in
If f is not differentiable then df is understood in the sense of distributions.
rn. n.
with a compact support and where du is the (Euclidean) Haar
For example,
r
Jrn. n
Ild!+lldu =
Vol av+ for compact domains V+
c
rn. n with C 1 -
boundary.
This can be seen with the approximation f. -> f +, E: -> 0, where f. ('U) = f + (u) for u E V+ and f.(u) = max(O, 1 - dist(u, V+)) outside V+. Hence the inequality (2) for !+ reduces to (1). On the other hand, the inequality (2) can be obtained by applying (1) to the regions Vt = {u E rn.nllf(u)1 :S t}, t ~ 0, and by a (clever) integration in t (see [Maz], [Bu.Ma.]).
The inequality (2) is a member of a large family of inequalities between various L p norms of f and df. For example, the Poincare inequality for functions f in a compact domain U
c
rn. n
with a smooth boundary claims
(3) where we assume Ju f du = 0 and where C(U) = (AdU))-1 for the eigenvalue Al of the Laplace operator -~ on U (with Neumann's boundary condition). The non-trivial content of
126
(3) is the implied inequality C(U) < 00 which amounts to '\1(U) > O. In fact, the inequality (3) can be obtained like (2) from an appropriate isoperimetric inequality for U. Namely, consider hypersurfaces Vo C U which have avo c au and which divide U into two regions U+ and U_ in U. Let I(Vo) = min(Vol U+, Vol U_)jVolVo and let Is(U) be the supremum of I(Vo) over all Vo C U. Then Cheeger's inequality (see [Bus]) claims >.t{U) ~ (2 I s(U)) -2 . (4)
£
which means
f2(u)du
~ 2 Is(U) fu lidfll 2du
(5)
for all functions f in U which have fu f(u)du = O. Thus (5) is proven like (2) by applying the isoperimetric inequality to the levels of the function f (see [eh]). The inequalities (1) and (2) were generalized by Brunn back in 1888 to an arbitrary ndimensional normed (Banach) space X = (X, II II). Namely, let the Haar measure dx in X be normalized to have the unit ball B = {x E X IlIxll ~ 1} of volume one (which disagrees with the Euclidean convention but has an advantage of simpler formulae) and let II II" denote the norm in the dual space X". THEOREM. (Brunn [Br]).
An arbitrary Cl-function f on X with a compact support
satisfies [
(
1 If(x)ln/(n-l) x
) (n-l)/n
dx ~
[
n- l 1x II df(x)lI"dx .
(6)
PROOF. (Knothe [Kn]) Fix a linear coordinate system x}, ••• , X n in X such that dx}, dx 2, ... , dX n = dx and let J,'(x) be a continuous function on X with a compact support SeX, whose interior is denoted by SO c S. LEMMA. There exists a Cl-map Y of SO into the cube C
= {O < Xi < 1} C
X with the
followt'ng two properties:
(1) The map Y is triangular: the i-th coordinate function of Y depends on
Xl, .. " Xi
only. That is
(e) The partial derivatives ~ are non-negative on SO and the Jacobian J(x)
= Det(;Y~) =
x,
PROOF. For s Ai(S)
IT ;Yi.
i=l
satisfies J(x)
X.
= J,'(x)j
[ J,'(x)dx,
1s
for all X
= (xt{s), ... ,xn(s)) E SO set
= {x = (Xl,."
,xn )
I Xj = Xj(s)
for
i
and
Xi
~ Xi(S)}
ESO .
127 and
B;(s) = {x = (Xl,'" ,Xn) I Xj = Xj(s) for j < i} . Then the map Y(s)
y;(s) =
= {y;(s)} with
!
Ai(S)
lL(s)dx;, ... , dx n /
J
Bi(s)
lL(s)dx;, ... , dXn , i = 1, ... , n
clearly satisfies (1) and (2).
o REMARK. The above formula equally applies to the characteristic function of an arbitrary open convex subset U in X which gives a triangular Cl-diffeomorphism of U onto C whose Jacobian identically equals (Vol U)-l. We are especially interested in the inverse of this diffeomorphism in case U is the open unit ball B = {llxll < 1} C X. This inverse map, called Y': C -+ B, clearly is triangular with the Jacobian ;: 1. We now compose the above map Y
= Yj<
with Y' and thus obtain a Cl-map
(Zl"",Zn)=Z=Y'oY: S°-+B which satisfies the above properties (1) and (2) as well as Y. This Z obviously has IIZ(X) II < 1 for all X E SO and the divergence div Z(x) = Ei..l ~(x) satisfies by the geometric-arithmetic mean inequality,
.
Finally, we take IL(X)
= I!(x) In/n-l
0= l div(I!(x)IZ(x))dx
and get
= ll!(x)ldiV
Z(x)dx + l
< dl!(x)l, Z(x) > dx ,
for the canonical bilinear pairing < , > between X' and X. Therefore,
j I!(x) In/n-ldx
=j
I!(x) IIL(x) l/ndx
~ n -1 (j I!(x) Idiv
Z(x)dxHj lL(x)dx) lin
~ n-l(j II d!(x)lI dxHj lL(x)dx)l/n Since the inequality (6) is homogeneous, we may normalize to
.
f lL(x)dx = 1, and then
(6)
follows from the above.
o REMARKS. (a) The inequality (6) generalizes with an obvious approximation argument = IR n , amounts to Sobolev's
to all functions! with (generalized) derivatives in L l which, for X inequality (2).
128 (b) The above proof shows that the equality in (6) holds if and only if f is a scalar multiple of the characteristic function of a metric ball in X. In fact, the proof gives an integral formula for the isoperimetric deficiency
(c) The inequality (6) (and its proof as well) obviously generalizes to the spaces X with non-symmetric norms. This generalization (in a slightly different but equivalent form) is called the Brunn-Minkowski inequality (see [Had]). (d) Brunn's inequality generalizes, up to a certain extent, to differential forms f on X of degree> 1, as was discovered (in the dual isoperimetric language by Federer and Fleming (see [F.F], [Gr2]). 2.2. Levy's Inequality. Let V be a closed n-dimensional Riemannian CCX>-manifold whose Ricci curvature is everywhere 2 (n - 1)R- 2 , R > 0, (which is the Ricci curvature of the
sn C rn. n+ 1 of radius R). Let V+ C V be a compact region with smooth boundary Va = av+ and let V'; be a round ball in the sphere sn of radius R in rn. n+ 1 such that Vol V';/Vol sn = Vol V+/Vol V.
round sphere
THEOREM. (Levy [Lev]).
the sphere Vo·
= av.;
The (n - l)-dimensional volume of Va is related to that of
by the inequality Vol Vo > Vol Vo· Vol V - Vol Sn
PROOF.
Consider the functional Vol ao+ on all domains 0+
C
V with a fixed n-
dimensional volume Vol 0+ = Vol V+. Then, by the global calculus of variations, there exists an extremal domain, say
n+
Unfortunately, the boundary
in V for which the (n - l)-volume
an+
an+
is the least possible.
is not necessarily smooth. However, a deep theorem of
F. Almgren [AI] claims the singularity to be negligible both from inside and outside (Notice that
n~ c
no
=
an+ an+
is smooth for n :S 7, see [Law].) of this
°
n+.
Furthermore, the non-singular locus
has constant mean curvature (this is obvious since the mean
curvature equals the normal derivative of the (n - l)-dimensional volume of n~). If this curvature, called Il, does not exceed that of Vo·, then the proof follows from lo5.B. Otherwise, we go to the complement
n_ = V\n+, in which (interior) direction the mean curvature of n~
equals -p. which is necessarily less than the mean curvature of Vo· in the direction of the ball
V~
= sn\v.;.
Then lo5.B applies to n~ and the proof is concluded.
o REMARKS AND COROLLARIES. (a) If V the classical isoperimetric inequality on
Among all domains in assumed by a round ball.
sn
sn:
= sn, then Levy's
inequality amounts to
with a fixed volume the minimal volume of the boundary is
129 A similar inequality holds true in the n-dimensional hyperbolic space Hn (see [Schm]), but for no space except m. n , sn and Hn one knows the exact solution of the isoperimetric problem. (b) Let Ve C V denote the e-neighbourhood of Vo in V. Then an obvious integration in e shows that Vol Ve Vol V/ ->--V 01 V - Vol sn . (c) Levy's inequality (and his proof as well) generalizes to all Riemannian manifold with a given (possibly negative) lower bound on the Ricci curvature (see [Grl]). This leads to sharp estimates on the eigenvalues Al ~ A2 ~ ... of the Laplace operator on V (see [B.G.M.J, [Ber]). (d) In order to apply Levy's inequality to a specific manifold V one needs some information on the Ricci curvature. In fact, the Ricci curvature is an easily computable invariant. EXAMPLES. (1) Let vn C m.n+I be a smooth hypersurface whose principal curvatures at some point Vo satisfy
XI, ••• ,X n
Then Ricci(v, v) ~ a for all v E Tv. (V). This follows from the famous theorema egregium of Gauss which expresses the intrinsic curvature of V in terms of the principal curvatures (see [G.K.M] or prove it yourself). (2) Let Riemannian manifolds Vi,"" Vk have Ricci(Vj ) ~ a, j = 1, ... , k. Then the Riemannian product V = VI X ••• X Vk also have Ricci(V) ~ a. This is immediate from the definition of the Riemannian structure g in V, which is g = gl EEl ... EEl gk. (3) Let V be the orthogonal group O(n) and let g be the (natural) left invariant metric on O(n) which is invariant under conjugations (which is equivalent to being right invariant as well as left invariant) and such that the circle consisting of the rotations around a fixed subspace m. n - 2 C m. n has length 271". (With these conditions g obviously is unique)then Ricci(O(n),g) ~ 1 everywhere (see [C.E.] for an explicit computation of the curvature of Lie groups).
APPENDIX II GAUSSIAN AND RADEMACHER AVERAGES
11.1. THEOREM: For all C < 00 and 2::; q < 00 there exists a constant K that if I3q(X) ::; C then for all n and Xl>-. _ ,X n E X
I I Lg,x, ,=1 n
L.{X)
::; K I L r,x, I n
,=1
= K(C,q)
such
L.{X)
In particular, Cq(X) ::; Kl3q(X) ((g')~=1 are independent symmetric gaussian variables normalized in L2' (r')~=1 are the Rademacher functions). Given a I-unconditional basis ZI, Z2, ••• in some Banach space X and a function m. -+ m., we may view f as acting on the finitely supported elements in X by applying f to each of the coefficients f:
11.2. DEFINITION: The p-convexity constant A p (resp. q-concavity constant B q) of the basis (z,) is the smallest A (resp. B) satisfying
A(Ellxn IIP)I/ p (EllxnllqP/q::; BII(EJxnJqP/qll)
II(ElxnJP)I/ PII ::; (resp.
for all finite sequences (x n ) of finitely supported elements in X. 11.3. DEFINITION: The upper p-estimate constant a p (resp. lower q-estimate constant bq ) of the basis (z,) is the smallest a (resp. b) satisfying
(resp.
IIEXnll ::; a(EllxnII P)I/p (EllxnIJqP/q::; bllExnlD
for all finite sequences (x n ) of finitely and disjointly supported (with respect to (z,)) elements inX. Clearly, ap
::;
A p , bp
::;
B p • Also for 1 ::; P ::;
00,
~
+ ~ = 1,
131
ap(Zi)
= bq(zi)
,
ap(zi)
= bq(Zi)
where (zi) are the biorthogonal functionals to (Zi) (see [L.T2]). Note also that a p :S ap(X), bq :S ,Bq(X). (a p , ,Bq are the gaussian type and cotype constants). 11.4. PROPOSITION: For aliI < r < q < B q :S K(r,q)b r
00
there exists a constant K(r,q) such that
for
1
00
and
Ap:S K(s,p)a. for
1 < p< s <
00 .
In particular B q :S K(r,q),Br(X)
for
1
< r < q < 00
A p :S K(s,p)a.(X)
for
1
and
PROOF: Let 1 < p < s < follows by duality.
00.
00 .
We shall show A p :S K(s,p)a., the result for B q, br
We need the following lemma, the proof of which we delay until after the proof of the proposition. 11.5. LEMMA: Let 1 < p < 8 < 00. There exist a 8equence 01 independent random variable8 (Ii) on some probability space (0,.1,1-') and a constant K = K(s,p) 8uch that
lor all finite sequences al,.· . , an.
Assume the lemma and let Xl, ... , X n be finitely supported elements in X
Xi
= LaijZj
.
j
For each W E 0 we define the function maxl~i~n Ixdi(W) I coordinatewise:
lx:!f<"" Ixdi(W) I = L mF laijli(w)IZj -
Note that for each
W
-
.
j
n
max Ixdi(W) I = LYi(W) l~.~n i=l where the Yi(W)-S are disjointly supported elements of X and IYi(W)1 :S Ixdi(w)I in the order induced by (Zi). It follows from the lemma that
132 By the I-unconditionality of (Zi) II(El x iI P)l/PIl
~ IIItYi(W)1I ~! IItYi(w)1I ~ a. !(EIIYi(w)II·)l/• •=1
~ a. !
.=1
(Elixill.lli(wW) 1/.
~K
a.(ElixiIlP)l/p
and
o 11.6. PROOF OF LEMMA U.S: Let (f,(w))~l be independent random variables with
where c is such that
f lidp, = 1.
Then
We conclude that, if E~=llaiIP = 1, then maxl$i$n ladil has the same distribution as It and in particular J maxl$i$n ladildp, = 1. This proves the right hand side equality. To prove the left hand side inequality let p < S < 00 and let (gi)~l be independent random variables such that
where d is such that
J g,dp, =
1. Then, by the first part of the proof
with K
= ( ! Ig,IP) lip < 00. o
11.7. Before proceeding to the proof of the Theorem we need another inequality which is a generalization due to Maurey of Khinchine's inequality.
133 THEOREM: for all C and q, 1::; C, q < 00, there exists a constant M = M(C,q) such that, If (zd is a 1-unconditional basis in some Banach space X with q concavity constant::; C, then
~1I(4= IXiJ2)1/211 ::; fol II L
• •
ri(t)xill ::;
(fol II L
ri(t)xill q
)l/
q
::;
MII(L IXiJ2)1/211 •
for all (x;) in X. PROOF: The left hand side inequality follows immediately from the triangle inequality and Khinchine's inequality in L 1 , the concavity assumption is not needed here. For the right hand side inequality:
(J II L ri(t)xill )l/ CiI(J I L ri(t)xiI )I/ Il q
q
::;
Q
Q
::; CKqll(L IXiI 2) 1/211 where K q is Khinchine's constant in L q •.
o
We turn now to the
11.8. PROOF OF THEOREM II.1: Let Xi = ri(t)xi in Lq(X). Since IIEri(s)xiIlL.(x) = IIEri(s)xiIlL.(L.(X» and IIEgi(s)xiIlL.(x) = IIEgi(s)xiIlL.(L.(X» and since Lq(X) has the same cotype q constant as X, we may assume we are dealing with (Xi) instead of (Xi)' In particular, without loss of generality, (Xi) is I-unconditional. Fix r > q, then, by Proposition 11.4, (Xi) is r-concave with r-concavity constant depending on q, rand Conly. Let (rij)i:'1 j~1 be a sequence of independent Rademacher functions. By Theorem 11.7, for all mEN,
By the central limit theorem, the sequence
tends in distribution to (gi)f=I' a sequence of independent, mean zero, gaussian random variables normalized in L 2 • Thus we get
o
APPENDIX III KAHANE'S INEQUALITY
We bring here a proof, essentially due to C. Borell, of Kahane's inequality 9.2. We begin with the Brunn-Minkowsky inequality (see also Appendix I).
mn •
111.1. THEOREM: Let A, B be two compact sets in
Vol(A
+ B)l/n
~ Vol(A)l/n
Then
+ Vol(B)l/n
.
PROOF: By a simple approximation procedure, we may assume that each of A and B is a union of finitely many disjoint sets, each of which is a product of intervals. The proof is by induction on the total number k of such rectangular boxed in A and B. IT k = 2, i.e. if A and B are rectangular boxes with sides of lengths (a,)i'::l and (b')~=l respectively, then
n
Vol(A
+ B) l/n = II (a, + b,) l/n,
n
Vol(A)
= II a:/ n ,
,=1
n
Vol (B)
i=l
= II b:/ n . i=l
By the inequality between the geometrical and arithmetical means,
n (
II
,= 1
ai
a' + b· ••
)
l/n
n (
b
i + II a' + b· i= 1 ••
)
l/n
1 n ai 1 n bi - + -n --=1. - n a' + b· a' + b·
<-
L
i= l '
•
L
i= l '
•
Assume now that A and B are composed of a total of k > 2 rectangular boxes and that the inequality holds for all sets A', B' such that A', B' are composed of a total of at most k - 1 boxes. We may and shall assume that the number of rectangular boxes in A is at least 2. Note that parallel shifts of A and B do not change the volume of A, B or A + B. We find such a shift of A with the property that one of the coordinate hyperplanes divides A in such a manner that at least one rectangular box in A is in each side of the plane. A is now divided into two sets A', A" each of which is a disjoint union of finite number of rectangular boxes and the number of boxes in each of A', A" is strictly smaller that the number in A. Now shift B parallel to the axes in such a manner that the same hyperplane divides B into B', B" with Vol B' _ Vol A' _ \
vorI1' - "VOTA - ".
Each of B', B" has at most the same number of rectangular boxes as B has.
135 By the induction hypothesis
Vol(A
+ B) 2: Vol(A' + B') + Vol (A" + B") 2: [(Vol A')l/n + (Vol B')l/njn + [(Vol A")l/n + (Vol B")l/njn = >.[(V 01 A)l/n + (Vol B)l/nt + (1 - >.)[(Vol A)l/n + (Vol B)l/nt = [(Vol A)l/n + (Vol B)l/njn o
Ill.2. It follows from Theorem III.1 that
Vol(>.A+ (1- >')B) 2: (Vol Af'(Vol B)l->' for all compact sets A, B in
rn. n
and all 0 < >. < 1. Indeed
V ol(>.A + (1 - >')B) lin 2: >.(V 01 A) lin
+ (1 -
If K is any convex body of finite volume in
>.)(Vol B) lin 2: {(Vol A)>' (V 01 B) 1->' pin.
rn. n , put JLK(A) =
V~~1~K). Then JLK clearly
satisfies the same kind of inequality
Ill.3.
JL(>.A JL(A)
Let JL be any Borel probability measure on mn satisfying (1 - >')B) 2: JL(A)>' JL(B) 1->'. Then for all symmetric convex sets A ~ mn with
THEOREM (C. Borell).
+
= 11 > !
JL((tA)C)~I1(I~I1)(1+t)/2
for all t>I.
PROOF. We have the inclusion
(check!). Consequently 1 - 11
= JL(A C) 2: JL((tA)c)2 / (t+ l )I1(t-l)/(t+l)
.
o 111.4. Let now (X,
II II)
be any normed space and let
where JL is the Lebesgue measure on the cube K Let
Xl, ••• , X n
= [-I,Ijn
normalized as to give JL(K)
n
A
= {a E Kj Ill: a;x;11 ;=1
E X. Assume
~ 3} .
= 1.
136 Then A is convex symmetric and JL(A) ~ Theorem 111.3 we get, for all t > 1,
111.4.1.
j, (since 3JL(A
JL{IIE?::laixili > 3t}
C
)
~
JK
IIEaixilldJL(a)
= 1).
Applying
~ ~(~)(1+t)/2
and consequently, for all p > 1,
for some constant K p depending only on p. It follows by homogeneity that
111.4.2. Kahane's inequality now easily follows from 111.4.2. Indeed, by the I-unconditionality of the Rademacher functions,
By the fact that (ad?::l and (!ai!ri(t))f=l have the same distribution and by the triangle inequality
We can now conclude from 111.4.2 that, for p > 1,
which proves Kahane's inequality. (The other side inequality is trivial.) We remark that this proof does not give the right order of magnitude for the growth rate of the constant. It only gives K p ~ Kp with K < 00 absolute. See [L.T2] for a proof which gives the right order of magnitude for the constant, K p
~
yP.
APPENDIX IV PROOF OF THE BEURLING-KATO THEOREM 14.4
Define
· StX - x A x= Il m - - t .... o t
whenever the limit exists. Then A is a linear (usually unbounded) closed operator with dense domain D(A). For any x E D(A) dStx dt
= lim
St+s x - St X
s.... o
s
exists and StX
= AStx = St Ax .
Let SUPt III - Stll = Po and let Po < p < 2, say p = STEP 1: For all f3 ~ p,
M
= M(po).
Indeed, since E~=o(
II 1-/, II :::;
e:)n = M. Also,
2+.p.
((f3 - 1)1 + St}-l exists and 11((f3 - 1)1 + St)-lll :::;
e:,
T
=
!ff-
with
E~=o( I-/,)n converges and its norm is at most
Thus,
STEP 2: There exists 0 < ()
larg
zl ~ I - (),
= (}(Po)
:::;
I
and M'
zl + A has a bounded inverse and
PROOF: Not~ that for.all t ~ 0 and x E D(A),
..
= M'(po)
such that for z E
138
+ ib. If one can choose t such that e- zt = 1 - {3 with {3 real and {3 2: p, then zt (e St - I) -1 exists and
Fix z = a
Choose t If {3
with
= fir.
Then,
= 1 + e-a'll"/Ibl, (J
then {3 2: p as long as small enough.
For such z and t we get
so that,
STEP 3: For
r
in the illustration
1* is small enough, Le., as long as larg zl > f -
(J
139 define, for e with larg
el < !'
!
!
Since larg zel ~ i - 8 + = i on the infinite rays, le-zel ~ e-a!z! for some positive ex so that the integral converges. One checks that it defines a holomorphic semigroup which extends St. To show that it extends St compute its derivative,
and we get that the two semigroups have the same generator. This of course means that they are equal. say,
To check that Se is a semigroup, notice that Se 2. Then,
r-
= 21r~ Jr' e-Ze(zI + A)-ldz for r' being,
By the resolvent formula,
and by the Cauchy formula,
SeSe'
= =
-1.
21rt
r e- ze - ze' (zI + A)-ldz
Jr
Se+e' .
Finally, we have to check that IISeli is bounded in the sector {Iarg to the reader (replace the unit circle with a larger one).
el < !}.
We leave this
APPENDIX V THE CONCENTRATION OF MEASURE PHENOMENON FOR GAUSSIAN VARIABLES
We bring here a simple proof, due to Maurey and Pisier, of Theorem V.l below which in turn is equivalent to Levy's Lemma 2.3 (up to modification of the constants). We also prove a proposition (VA) relating the deviation from the mean to the deviation from the median.
Let F: IR n -+ IR be a Lipschitz function with constant (J (IR n is endowed with the euclidean metric). Let gl, ... ,gn be independent, mean zero, normalized in L 2 , gaussian variables. Then,
V.I.
THEOREM:
PROOF:
We may and shall assume that F is continuously differentiable.
Let
H = (hi' ... ' h n ) be another sequence with the same distribution as G = (gb ... , gn) and independent of it. For each 0 ~ (J ~ i put Gs = G sin (J+H cos (J. Then, by the invariance of the gaussian distribution under orthogonal transformations, Gs and fsGs = G cos (J-H sin (J have the same joint distribution as G and H. Consequently, for any convex function
E
= E
11:/2 1
0
f1l:/2
= E
d
d(JF(Gs)d(J)
d
(grad F(Gs), d(JGs)d(J)
2111:/2 0 E
~:;
11"
= E
For any ,X E ill. we get •
'
11"
n aP
.
E exp('x(F(G} - EF(·))) ~ E eXP('x2~ aXi (G) . hi) . • =1
Integrating 'first with respect to the hi and using the fact that
d
d(JGs))d(J
141 we get A21r2
E exp(A(F(G) - EF(.))) ::; E exp (
8
L n
aF
_(G)2
;=1
)
ax;
The rest of the proof is now standard.
o V.2. Next we show how to deduce from Theorem V.l a similar theorem for sn-1. COROLLARY: Let f: sn-1
-+
IR be a function with Lipschitz constant u. Then
where I-' is the Haar measure on sn-1 and b > 0 an absolute constant. PROOF: First notice that, by the invariance of the canonical gaussian measure on IR n under orthogonal transformations,
has the same distribution as a
i:
Given a function f:
= (a1"'"
sn-1
-+
an) does with respect to dl-'(a).
IR with Lipschitz constant u and
IR -+ IR by i(x) = IIxllf( ~), where Lipschitz constant of is at most 3u. n
i
Ilxll =
J fdl-' = 0,
define
(E xn L Then it is easily checked that the
Now, for any b > 0,
1-'(1/1 > G) = p(l/( 11:11 ) I > G) = p(llglr1Ii(g)1 > C) = p(\i(g) I > GIIglD ::; p(li(g) I > aGy'n) + P(lIgll < a.fii) ::; p(li(g)/ > aGy'n) + p(\lIgll- Ellglll > Ellgll- av'n). A simple computation shows that Eligil > 2byn for some b > 0 absolute. For that b we get from Theorem V.l, applied to
i
and to 11·11, that
Since we may assume G ::; u, we get that the first term is dominating and the result follows.
o
142 V.3. REMARK: The last corollary evaluates the probability of a large deviation of
I from
its expectation while the results in Section 2 deals with deviation from the median. This is another advantage of the method here. It saves us the trouble of passing from the median to the average (Lemma 5.1). Also, as we shall see below, it is not hard to get, using Theorem V.1, a similar estimate for the deviation from the median. V.4. We conclude this appendix with a proposition showing that estimation of large deviation
from the median is equivalent to the estimation of large deviation from the expectation.
PROPOSITION: Each of the following four conditions implies the other three, where the constants (Ki,8i ) depend linearly each on the other. 1.
3A s.t. \:IC
2.
\:IC
where 3.
\:IC
..j.
\:IC
I and 1 are independent and identically distributed.
(M I is the median of J). More precisely,
and
83 8z 81 81 > 8 > - > - > - 4 - 4 - 8 - 32
.
PROOF: 1 => 2 -
P(II - II> C)
~
C
-
C
P(II - AI> 2) + P(II - AI> 2)
~ 2K 1e- 01C'j4
2=>3 E exp(>,zll -
liZ) = ~
L XJ
2>,zCe>.'c' P(II -
2Kz >'Z
10
00
11> C)dC
CeA'C'-o,c' dC .
143
For A =
/Fi, we get Eexp(821/;iI2)
~K2.
Now,
and
3 =* 4 is similar to the end of the proof 7.5: Let Co
P(ll - Ell> Co)
~
= V'Og6;K3.
Then
I
2.
In particular,
P(f> Co + EJ), P(f < EI - Co)
~
I
2
so that
EI - Co < M I < Co + EI . Now, for C
~
2Co,
P(II - MIl> C)
~
P(II - Ell> C - Co)
~
K ae-
63
(C-CO ) '
~ Kae-!j.c· . For C
< 2Co,
and
so that 4 holds with 84
=~
and
K4 = 2Ka.
4 =* I is trivial.
o
NOTES AND REMARKS
2.1 The theorem is stated in Levy's [Lev] with a proof which has not been understood for a long time. A complete proof appears in [Schm]. In Appendix I a proof (of Gromov) is given which is based on Levy's original idea. See also [F.L.M.] for a short proof. 2.3 is from [Lev]. 2.4-2.8 are from [Ml].
3.1 There are a substantial number of results about distances between different finite dimensional normed spaces. A detailed study of these problems can be found in [T-H]. We only mention a few results. Let Kn be the family of all n-dimensional normed spaces equipped with the Banach-Mazur distance. Theorem 3.3 implies that the diameter of Kn is :::; n. It was shown by E. Gluskin [Gil that diam Kn ~ en for some absolute constant e > O. However,. when restricted to some natural subclasses, the diameter is much smalier. For example, N. Tomczak-Jaegermann [T-J2] has proved that the distance between any two spaces with 1symmetric bases in Kn is at most Cyn for some absolute C < 00. Recently J. Lindenstrauss and A. Szankowski [L.S.] have shown that for some (specific) Q < 1 and C < 00 the distance between any two members of Kn with I-unconditional bases is at most Cn"'. (The Q computed in [L.S.] is ~ while one naturally expects the right Q to be or + c with C = C(c).) A similar estimate was obtained in [B.M.l] for the distance between X E Kn and its dual, X'. In another direction, subfamilies of Kn with some conditions on their type and cotype were considered in [Kwl], [F.L.M.], [T-J.3], [D.M.T-J.] and [B.G.] where again estimates of order better than n were given to their diameters. Theorem 3.3 also implies that the radius of Kn is equal to yn with e~ in (one of?) the center(s). If X E K n is of almost extreme distance from e~, Le., d(e~,X) ~ en (e > 0 fixed), then it was proved in [M.W.] that X necessarily contains a (1 + c)-isomorphic copy of with k = k(e, c, n) -+ 00 as n -+ 00.
!'
! !
et
3.3 is taken from F. John's [Jo]. 3.4 This is a weak form of the Dvoretzky-Rogers Lemma [D.R.] (or see [Da]). The estimate
can be improved to 1 Johnson.
~ IIxili ~ Jj.~t~):;:J"
The proof given here was shown to us by W.
4.2, 4.3 were proved in [Ml]. The estimates on the constant e(c) in 4.2 Remark a, and 4.3,
145 can be improved-to C(e) ~ Ce 2. This follows from a recent proof of Dvoretzky's Theorem due to Y. Gordon [Go]. 4;.4-4.7 are proved in [F.L.M.]. 4.8-4.11 are taken from [M3]. The best constant in 4.8 is
";f! Ixl ~ IIxll (see P.T-J.]).
5.1-5.4 are taken from [F.L.M.] except for Fact 2 of 5.4 which was proved in [B.D.G.J.N.].
The condition a· b ~
vn in 5.1 is not needed (T. Figiel, private communication).
Actually
all of Lemma 5.1 is not needed, instead one can use Proposition VA. 5.3 admits a lot of generalizations, we indicate here only one direction: proved that for any A < 1, t~ contains a C(A)-isomorphic copy of t~, for k
Kashin [Kas] has
= [An].
The proof
uses a different approach involving the following property of t~: Let 0 be the ellipsoid of maximal volume inscribed in the unit ball, K, of t~, then vr(K) = (Vol KIVol O)l/n is bounded independently of n (see also [Sz2]). The quantity vr(K), called the volume ratio of K, was introduced and studied for various K's in [S.T-J.], where a generalization of Kashin's theorem is proved for spaces whose unit balls have bounded volume ratio. For recent developments on this subject (in particular its relation with cotype 2), see [B.M.2] and [MLP.]. For the original proof of Khinchine's inequality see [L.T.1]. The best constants = 1, by S. Szarek [Szl] (AI = ~), and for all the p's by U. Haagerup [Haa]. 5.5
Ap,Bp were computed, for p
5.7 was first proved in [M1]. There is a disturbing gap between k(t~) ~ log n and k(t~) ~
n 2 / q (which becomes a constant for q = log n for which d(t~,t~) = e). This gap can be bridged - one can prove k(t~) ~ cqn 2 / q for all q ~ log n with an absolute c > O. This estimate is best possible up to numerical constants. 5.8 The original proof of Dvoretzky's Theorem only gives k(X) ~ cy'log nllog log n. As stated here, the theorem was first proved in [M1l (with the proof given here). 5.9 The theorem (with a different dependence on A) was proved in [M2]. We follow [M3]. 6.1 The notion of Levy family was introduced in [G.M.] which also contains a list of examples
of Levy families and applications of the concentration phenomenon of such families to topology and fixed point theorems. 6.2 was observed and used in [A.M.]. 6.3 was proved and used for the Local Theory by Maurey [Ma]. 6.5-6.7 These examples were observed in [G.M.]. Concentration properties for Wn,k and
Gn,k were also noted and applied in [M4], [M5]. 6.9 was proved in [G.M.]. This method was later extended in [Al.M.] to discrete cases.
146
1.4 is due to K. Azuma (see [St] for this and related inequalities). For other related inequalities, see Lemma 8.4 below and Proposition 3.1 in [J.S.Z.]. 1.5 was proved by B. Maurey [Maj. The proof here follows [Sell. We remark that unlike the
similar case of
E'2
no precise isoperimetric inequality for JI n is known.
1.1-1.8 is taken from [SCI]. 1.9 For a proof through an isoperimetric inequality, see L.H. Harper's [Har] (and [A.M.1] for
the form in which it appears here). [Fr.F] contains a simpler proof of Harper's isoperimetric inequality. 1.15 is taken from [Sc2], which contains also generalizations to spaces other than l~. 8.1 p-stables were introduced by P. Levy. Their relevance to the problem of embedding lp in L r was noticed by M.1. Kadec. 8.2-8.8 is taken from [J.S.] in which a more general case is also considered (in the statement
of Theorem 8.8 one can replace l~ by l~ for any 0 < r :5 1 and also by 1 < r < p provided 1 + e is replaced with Kr,p and f3 = f3(r,p)). Theorem 8.8 was further generalized, replacing l~ by a wide family of spaces, by G. Pisier [Pis2], see Chapter 13. For generalizations in a different direction, replacing l;' by a general finite dimensional subspace of L p see [Sc3], [Sc4]. 8.4 This inequality is due to G. Pisier. 9.1 The type and cotype inequalities were first considered by J. Hoffmann-JorgenseI]- [H-J)
in connection with limit theorems for independent Banach space valued random vatiables. [M.P.] contains the first major relations between these inequalities and geometrical properties of Banach spaces. 9.2 is proved in [Kah]. The original proof as well as the proof in Appendix III do not give the right order of the constant K p • For a proof, due to Kwapien, which gives the right order
Kp
~
yIP see [Kw2] or [L.T.2J.
9.6 is taken from [F.L.M.]. 9.1 appears here for the first time. It should be compared with Theorem 13.12 below. 9.10 The Rademacher projection, Rad n , and its relevance to duality between type and cotype
was noticed in [M.P.]. 10.
is based on [A.M.1] and [A.M.2J. However, the main idea of the proof of Theorem 10.7
is an adaptation of Krivine's original proof [Kr] to the finite dimensional case. 11.
A good reference for combinatorial treatment of Ramsey's Theorem is [G.R.S.]. For
a review of applications of Ramsey's Theorem to Banach space theory see Odell's rOd]. A.
147 Brunei and L. Sucheston were the first to notice the relevance of Ramsey's Theorem to Banach space theory; they proved Theorem 11.4 (cf.[B.S.l1, [B.S.2]). 11.8 The Lemma here is due to James [Ja]. 12.3-12.4 are due to J.L. Krivine [Kr]. We follow here a simpler proof due to H. Lemberg
[Lem]. The proof given for 12.5 appears here for the first time. It simplifies the proof of the Maurey-Pisier Theorem 13.2. 13.2 is due to Maurey and Pisier [M.P.]. We follow [M.S.] (for the cotype case, 13.8-13.9) and [Pis2] (for the type case, 13.10-13.12). [Sc3] and [Sc4] contain some extensions of 13.12: It is proved there that any k-dimensional subspace of L p , 1 < p < 2, (1 + c) imbeds into X provided k1+l/P(log k)-l ~ S(p,c)(STp(X))q. It is not known if the conclusion holds with
k
~
S(p,c)(STp(X))q.
14.5 is contained in [Pisl]. 14.6 is contained in [Pis4] and [PisS] with a different proof. The proof here was shown to us
by T. Figiel. The estimate IIRad n XIl ~ K log n for X with dim X = n is best possible, up to the choice of K. This was shown by J. Bourgain [Bou]. But in special cases (notably, spaces with I-unconditional bases) one gets IIRad n X11 ~ 15.
Kvrogn,
see [PisS].
The material in this chapter is due mainly to T. Figiel and N. Tomczak-Jaegermann
[F.T-J.]. 15.2 is a generalization, due to D. Lewis [Lew], of a theorem of F. John [Jo].
l.l.5.B first appear in that form in [Grl]. It is more commonly referred to as the LevyGromov Theorem. 11.1 The theorem is due to Maurey and Pisier [M.P.]. The proof here is different. 11.2-11.3 For more information about p-convexity, q-concavity, upper p-estimate and lower
q-estimate, see [L.T.2]. 111.3 is from [Bo]. 111.4 G. Pisier showed us how to deduce Kahane's inequality from Borell's theorem.
IV. The proof is taken from [R.S.].
148 V.I See [Pis7] for this and some generalizations.
V.4 is probably known to several people but we don't know of any reference.
INDEX (Roman numerals refer to appendices) Hyperbolic plane 1.1.1
Approximate eigenvector, eigenvalue 12.1; 12.2, 12.3, 12.5 Banach limit 12.2 Banach-Mazur distance 3.1 Bernstein's inequality 14.6 Beurling-Kato theorem 14.4, IV Block sequence 10.4-10.7 Borell's theorem 111.3 Brunn-Minkowski inequality 1.2.1, III.1 Brunn's theorem 1.2.1 Cheeger's inequality 1.2.1 Concavity (q-) 11.2, 11.7, 11.8 Concentration function 6.1 Concentration of measure 2.3 Convexity (p-) 11.2 Conditional expectation 7.1, 7.2 Cotype 9.1, 9.3, 9.6, 9.10, 9.11, 10.3, 10.4, 12.5, 13.1, 13.2, 13.3, 13.5, 14.7, 15.4, Dual norm 3.1.1, 15.1, 15.2 Dvoretzky's theorem 4.1, 5.8, 13.8, 13.12 Dvoretzky-Rogers theorem 3.4, 5.8 E~ 6.2, 7.9 Ellipsoid of maximal volume 3.2, 3.3, 3.4, 5.8,9.6 Exponential map 1.1.5
Finitely representable (K, crudely, block) 11.6-11.8, 12.3-12.6, 13.2, 13.12, 14.2, 14.3, 14.5 g-length 1.1.1 Gauss' theorema egregium 1.2.2 Gauss-Wei! formula 1.1.3, 1.1.4 Gaussian cotype 9.4 Gaussian type 9.4, 9.7 Gaussian variable 9.4,'9'.7, 15.1,11.1; 11.8, V~
Grassman manifold 1.5, 6.7.2 Haar measure 1.1 Homogeneous space 1.2, 1.3, 6.6
.
Ideal norm 15.1 Invariant riemannian metric 1.1.2 Isomorphic (A-) 3.1 Isoperimetric inequality 2.1, 1.2 F. John's Theorem 3.3, 15.2
k(X), k(X, e) 4.3-4.7,5.8,9.6,9.7 k(l~)
k(l~)
5.3 5.4
k(lex» 5.7 Kahane's inequality 9.2, 9.4.3, III Khinchine's inequality 5.5, 7.14 Krivine's theorem 10.5, Ch. 12, 13.8 i-norm 15.1
l; spaces 5.2, 10.5-10.7, Ch.
12, 13.2, 13.12, 13.13, 13.16 L 2 (X) 9.8,9.10 Laplacian 6.8 Length (of a metric probability space) 7.7,7.8, 10.6 Levy family 10.0 Levy family, normal 6.1-6.8, 7.5, 7.9, 10.3 Levy's lemma 2.3, V.1 Levy mean 2.3 Levy's theorem 1.1.4.A, 1.1.5.B Lower q-estimate 11.3 Marriage theorem 1.1 Martingale 7.3, 7.6, 8.4, 9.7 Martingale difference 9.7 Maurey-Khinchine inequality 11.7 Maurey-Pisier theorem Ch. 13 Median 2.3, 4.1, 4.8, 5.1, 10.1, 10.3.1, V.3, V.4 Mean curvature 1.1.4 OrthogonaLgroup 0;.. 1.2, 1.5, 6.3 lIn 6.3, 7.5, 7.7 p-stable (symmmetric) 8.1, 8.2, 13.10, 13.11, 13.15 Plane, tangent, normal 6.8
150 Poincare inequality 1.2.1 Poincare metric 1.1.1 r-norm 15.1 Rad n , RadnX 9.10, 9.11, 13.16, Ch. 14, 15.3, 15.4 Rademacher functions 5.5, 7.14, 9.0, 9.4, 9.9, 9.10, 15.1, ILl, II.8 Ramsey's theorem 11.1, 11.2 Ricci curvature 6.4, 1.1.5, 1.2.2 Riemannian manifold 6.3, 6.8, 6.9, App. Riemannian metric 1.1.1 Riemannian metric, induced 1.1.2 Riemannian structure 1.1.1 1.5 SOn 6.5.1, 6.6
sn-l
Semigroup, analytic 14.1, 14.4, IV Sobolev's inequality 1.2.1 Spreading 11.7, 11.8, 11.9, 12.3, 12.5, 12.6, 13.8 Stable type 13.10, 13.12 Stiefel manifold 1.5,6.7.1 Strongly continuous (semigroup) 14.4 Symmetric sequence 10.2, 10.3, 10.7 Type 9.1, 9.3, 9.4, 9.10, 9.11, 13.1, 13.2, 13.5, 13.10, 14.7, 15.10 Unconditional sequence 10.2, 10.4, 11.8, 11.10, 12.3, 12.5, 12.6, 13.8 Upper p-estimate II.3 Walsh system 9.9, 14.1 Weak lp-norm 8.3
REFERENCES
[AI]
F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. A.M.S., 4 (1976).
[Al.M.]
N. Alon and V.D. Milman, Ai> isoperimetric inequalities for graphs, and superconcentrators, J. Comb. Theory, Ser. B, 38 (1985), 73-88.
[A.M.1]
D. Amir and V.D. Milman, Unconditional and symmetric sets in n-dimensional normed spaces, Israel J. Math., 37 (1980),3-20.
[A.M.2]
D. Amir and V.D. Milman, A quantitative finite-dimensional Krivine theorem, Israel J. Math., 50 (1985), 1-12.
[B.D.G.J.N.j G. Bennett, L.E. Dor, V. Goodman, W.B. Johnson and C.M. Newman, On uncomplemented subspaces of L p ,l < p < 2, Israel J. Math., 26 (1977), 178187. [B.G.]
Y. Benyamini and Y. Gordon, Random factorization of operators between Banach spaces, J. d'Analyse Math., 39 (1981),45-74.
[B.G.M.]
M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Varillte Riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer 1971.
[Ber]
P.M. Berard, Lectures on spectral geometry, IMPA 1985, Rio de Janeiro. Lecture Notes in Mathematics, Vol. 1207, Springer 1986.
[Be]
A. Beurling, On analytic extension of semigroups of operators, J. Funct. Anal., 6 (1970),387-400.
[Bo]
C. Borell, The Brunn-Minkowski inequality in Gauss spaces, Inventiones Math., 30 (1975), 207-216.
[Bou]
J. Bourgain, On martingales transforms in finite dimensional lattices with an appendix on the K-convexity constant, Math. Nachrichten, 119 (1984), 41-53.
[B.M.1]
J. Bourgain and V.D. Milman, Distances between normed spaces, their subspaces and quotient spaces, Integral Equations & Operator Theory, 9 (1986), 31-46.
[B.M.2]
J. Bourgain and V.D. Milman, On Mahler's conjecture on the volume of a convex symmetric body and its polar, Preprint, IRES 1985.
[B.S.1]
A. Brunei and L. Sucheston, On B-convex Banach spaces, Math. Systems Th., 7 (1974), 294-299.
[B.S.2]
A. Brunei and L. Sucheston, On J-convexity and some ergodic super-properties of Banach spaces, Trans. A.M.S., 204 (1975),79-90.
[Br]
R. Brunn, Uber Ovale und Eiflachen, Inaug. Diss., Miinchen, 1887.
[Bu.Ma.]
Yu. Burago, and V. Mazya, Some problems of the potential theory and the function theory for domains with non-regular boundary, Zapiski LOMI, Leningrad, 1967 (Russian).
h:,
[Bus]
P. Buser, On Cheeger's inequality Al ~
[Ch]
J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, Symp. in honor of Bochner, Princeton, 1970, 195-199.
Preprint.
152 [C.E.]
J. Cheeger and E.G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975.
[D.M.T-J.]
W.J. Davis, V.D. Milman and N. Tomczak-Jaegermann, The distance between certain n-dimensional spaces, Israel J. Math., 39 (1981), 1-15.
[Da]
M.M. Day, Normed Linear Spaces, Springer, 1973.
[Do]
W.F. Donoghue Jr., Distributions and Fourier Transforms, Academic Press, 1969.
[D.S.]
N. Dunford and J.T. Schwartz, Linear Operators, Part I, Wiley, 1957.
[Dv.]
A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Symp. on Linear Spaces, Jerusalem 1961, 123-160.
[D.R.]
A. Dvoretzky and C.A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci., U.S.A., 36 (1950), 192-197.
[F.F.]
H. Federer and W. Fleming, Normal and integral currents, Ann. of Math., 72 (1960),458-520.
[Fe]
W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley, 1966.
[Fi]
T. Figiel, A short proof of Dvoretzky's theorem on almost spherical sections, Compositio Math., 33 (1976), 297-301.
[F.L.M.]
T. Figiel, J. Lindenstrauss and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math., 139 (1977), 53-94.
[F.T-J.]
T. Figiel and N. Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math., 33 (1979), 155-171.
[Fr.F.]
P. Frankl and Z. Fiiredi, A short proof for a theorem of Harper about Hammingspheres, Discrete Math., 34 (1981), 311-313.
[Gi]
D.P. Giesy, On a convexity condition in normed linear spaces, Trans. A.M.S., 125 (1966), 114-146.
[Gil
E.D. Gluskin, The diameter of the Minkowski compactum is roughly equal to n, Funct. Anal. Appl., 15 (1981), 72-73.
[Go]
Y. Gordon, Some inequalities for gaussian processes and applications, Israel J. Math., 50 (1985),265-289.
[G.R.S.]
R.L. Graham, B.L. Rothschild and J.H. Spencer, Ramsey Theory, Wiley, New York,1980.
[G.K.M.]
D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie in Gruppen, Springer, 1968.
[Grl]
M. Gromov, Paul Levy's isoperimetric inequality, Prepri-nt, 1980.
[Gr2]
M. Gromov, Filling Riemannian manifolds, J. Diff. Geom., 18 (1983), 1-147.
[Gr3]
M. Gromov, Partial Differential Relations, Springer Verlag, to appear.
[G.M.]
M. Gromov and YD. Milman, A topological application of the isoperimetric inequality, Amer. J. Math., 105 (1983), 843-854.
[Haa]
Uffe Haagerup, The best constants in the Khinchine inequality, Studia Math., 70 (1982), 231-283.
153 [Had]
H. Hadwiger, Vorlesungen iiber Inhalt, OberfHi.che und Isoperimetrie, Springer Verlag, 1957.
[Har]
L.H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Comb. Theory, 1 (1966), 385-393.
[H-J]
J. Hoffmann-Jorgensen, Sums of independent Banach space valued random variables, Aarhus Universitat, 1972/73.
[Ja]
R.C. James, Uniformly non-square Banach spaces, Ann. of Math., 80 (1964), 542-550.
[Jo]
F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, Interscience, New York, 1948, 187-204.'
[J.S.]
W.B. Johnson and C. Schechtman, Embedding (1982), 71-85.
[J.S.z.]
W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Prob., 13 (1985),234-253.
[Rah]
J.P. Kahane, Series of Random Functions, Heath Math. Monographs, Lexington, Mass., Heath & Co., 1968.
[Kas]
B.S. Kashin, Sections of some finite dimensional sets and classes of smooth functions, Izv. ANSSSR, ser. mat. 41 (1977), 334-351 (Russian).
[Ka]
T. Kato, A characterization of holomorphic semi-groups, Proc. A.M.S., 25 (1970), 495-498.
[Kn]
H. Knothe, Contributions to the theory of convex bodies, Michigan Math. J., 4 (1957),39-52.
[Kr]
J.1. Krivine, Sous-espaces de dimension finie des espaces de Banach reticules, Ann. of Math., 104 (1976), 1-29.
[Kwl]
S. Kwapien, Isomorphic characterizations of inner product spaces by orthogonal series with vector coefficients, Studia Math., 44 (1972),583-595.
[Kw2]
S. Kwapien, A theorem on the Rademacher series with vector coefficients, Proc. Int. Conf. on Prob. in Banach Spaces, Lecture Notes in Math. Vol. 526, Springer, 1976.
[Law]
B. Lawson, Minimal varieties, Proc. of Symp. in Pure Math., A.M.S., 27, Part I, (1975), 143-177.
[Lem]
H. Lemberg, Nouvelle demonstration d'un theoreme de J.L. Krivine sur la finie representation de t p dans un espace de Banach, Israel J. Math., 39 (1981), 341-348.
[L.W.Z.]
R. Lepage, M. Woodroofe and J. Zinn, Convergence to a stable distribution via order statistics, Ann. Prob.,9 (1981),624-632.
[Le]
P. Levy, Problemes concrets d'analyse fonctionnelle, Gauthier Villars, Paris 1951.
[Lew]
D.R. Lewis, Ellipsoids defined by Banach ideal norms, Mathematika 26 (1979), 18-29.
[L.S.]
J. Lindenstrauss and A. Szankowski, On the Banach-Mazur distance between
t;'
into
tl ,
Acta Math., 149
154 spaces having an unconditional basis, Aspects of Positivity in Functional Analysis, North Holland, to appear.
[L.T.]
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics Vol. 338, Springer-Verlag, Berlin, 1973.
[L.T.1.]
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin, 1977.
[L.T.2.]
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer-Verlag, Berlin, 1979.
[Lo]
M. Loewe, Probability Theory I, Springer-Verlag, 1977.
[Mar.P.]
M.B. Marcus and G. Pisier, Characterizations of almost surely continuous pstable random Fourier series and strongly stationary processes, Acta Math., 152 (1984),245-301.
[Mal
B. Maurey, Construction de suites symetriques, C.R.A.S., Paris, 288 (1979), 679-681.
[M.P.]
B. Maurey and G. Pisier, Series de variables aleatoires ve<;torielles independentes et proprietes geometriques des espaces de Banach, Studia Math., 58 (1976), 45-90.
[Maz]
V. Mazya, Classes of domains and imbedding theorems for function spaces. Dokl. Ak. Sci. USSR 133 #3, 527-530 (1960), (English Transl. p. 882-884).
[Ml]
V.D. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Func. Anal. Appl., 5 (1971),28-37 (translated from Russian).
[M2]
V.D. Milman, Almost Euclidean quotient spaces of subspaces of finite dimensional normed spaces, Proc. A.M.S.,94 (1985),445-449.
[M3]
V.D. Milman, Random subspaces of proportional dimension of finite dimensional normed spaces: Approach through the isoperimetric inequality, Banach Spaces, Proc. Missouri Conference 1984, Lecture Notes in Mathematics, Vol. 1166, Springer 1985.
[M4]
V.D. Milman, Asymptotic properties of functions of several variables defined on homogeneous spaces, Soviet Math. Dokl., 12 (1971), 1277-1281.
[M5]
V.D. Milman, On a property of functions defined on infinite-dimensional manifolds, Soviet Math. Dokl., 12 (1971), 1487-1491.
[MLP.]
V.D. Milman and G. Pisier, Banach spaces with a weak cotype 2 property, Israel J., to appear.
[M.S.]
V.D. Milman and M. Sharir, A new proof of the Maurey-Pisier theorem, Israel J. Math., 33 (1979),73-87.
[M.W.]
V.D. Milman and H. Wolfwon, Minkowski spaces with extremal distance from Euclidean space, Israel J. Math., 29 (1978), 113-130.
[Mi]
J. Milnor, Morse Theory, Annals of Math. Studies No. 51, 1963, Princeton University Press.
[Od]
E. Odell, Applications of Ramsey theorems to Banach space theory, Notes In Banach Spaces, University of Texas Press, Austin and London, 1980.
155 [P.T-J.]
A. Pajor and N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces, preprint.
[Pel
A. Pelczynski, Geometry of finite dimensional Banach spaces and operators ideals, Notes in Banach Spaces, University of Texas Press, Austin and London, 1980.
[Pie]
A. Pietsch, Operators Ideals, North Holland, 1978.
[Pis1]
G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. Math., 115 (1982), 375-392.
[Pis2]
G. Pisier, On the dimension of the l;-subspaces of Banach spaces, for 1 Trans. A.M.S., 276 (1983),201-211.
[Pis3]
G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math., 151 (1983), 181-208.
[Pis4]
G. Pisier, Un theoreme sur les operateurs lineaires entre espaces de Banach qui se factorisent par un espace de Hilbert, Ann.Scient.Ec.Norm. Sup., 13 (1980), 23-43.
[Pis5]
G. Pisier, Remarques sur un resultat non publie de B. Maurey, Sem. d'Ana!. Fonctionnelle 1980/81, Ecole Polytechnique, Paris.
[Pis6]
G. Pisier, Factorization of Linear Operators and the Geometry of Banach Spaces, CBMS Regional Conf. Series in Math., 1986.
[Pis7]
G. Pisier, Probabilistic methods in the geometry of Banach spaces, preprint.
[Ra]
F.P. Ramsey, On a problem of formal logic, Proc. Lond. Math. Soc. (2),30 (1929), 264-286.
[R.S.]
M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Analysis, Self Adjointness, Academic Press, 1975.
[ScI]
G. Schechtman, Levy type inequality for a class of metric spaces, Martingale Theory in Harmonic Analysis and Banach Spaces, Springer-Verlag 1981, 211215.
[Sc2]
G. Schechtman, Random ePlbedding of euclidean spaces in sequence spaces, Israel J. Math., 40 (1981), 187-192.
[Sc3]
~
p
< 2,
G. Schechtman, Fine embeddings of finite dimensional subspaces of L p , 1 < 2, into l"{', Proc. A.M.S.,94 (1985), 617-623.
~
p
[Sc4]
G. Schechtman, Fine embeddings of finite dimensional subspaces of L p , 1 ~ < 2, into finite dimensional normed spaces II, Longhorn Notes, Univ. of Texas, 1984/85. p
Die Brunn-Minkowski Ungleichung,
Math.
Nachr., 1 (1948),
[Schm]
E. Schmidt, 81-157.
[St]
W.F. Stout, Almost Sure Convergence, Academic Press, 1974.
[Szl]
S.J. Szarek, On the best constant in the Khinchine inequality, Studia Math. 58 (1976), 197-208.
[Sz2]
S.J. Szarek, On Kashin almost Euclidean orthogonal decomposition of l~, Bull. Acad. Polon. Sci. 26 (1978),691-694.
156 [Szan]
A. Szankowski, On Dvoretzky's theorem on almost spherical sections of convex bodies, Israel J. Math., 17 (1974), 325-338.
[T-J.1]
N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite Dimensional Operator Ideals, Pitman, To appear.
[T-J.2]
N. Tomczak-Jaegermann, The Banach-Mazur distance between symmetric spaces, Israel J. Math., 46 (1983), 40-66.
[T-J.3]
N. Tomczak-Jaegermann, The weak distance between finite-dimensional Banach spaces, Math. Nach., 119 (1984),291-307.
[Vi]
N. Ja. Vilenkin, Special functions and the Theory of Group Representations, Translations of Math. Monographs, A.M.S., Vol. 22 (1968).
[Zy]
A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1977.
