ON A CONJECTURE OF LINDENSTRAUSS* BY VICTOR KLEE ABSTRACT It is proved that each n-dimensional centrally symmetric convex polyhedron admits a 2-dimensional central section having at least 2n vertices. Some other related results are obtained and some unsolved problems are mentioned.
The following conjecture of Joram Lindenstrauss was transmitted to me by Branko Griinbaum: (E) I f P is an n-dimensional centrally symmetric convex polyhedron, then some 2-dimensional central section of P has at least 2n vertices. I present here a proof of this conjecture as well as a dual statement (I) about affine images of polyhedra. In fact, the simplest approach to (Z) seems to be by way of (I), using the duality between sections and affine images as in [2, 3]. Let us begin with the proof of (I), where of course (Z) and (I) both require that n>2. (I) If P is an n-dimensional centrally symmetric convex polyhedron, then some 2-dimensional affine image of P has at least 2n vertices. Proof. The assertion is obvious for n = 2. Suppose it is known for n = k and consider the case n = k + 1, where we assume without loss of generality that P lies in a (k + 1)-dimensional real vector space E and is centered at the origin 0 of E. We wish to show that some 2-dimensional linear image of P has at least 2k + 2 vertices. Let V be the set of all vertices of P and let ~ be a linear transformation of E onto a k-dimensional vector space F such that ~ is biunique on V. Then ~P is a k-dimensional centrally symmetric convex polyhedron, so (by the inductive hypothesis) there exists a linear transformation ~/' of F onto a 2-dimensional vector space G such that the convex polygon .Q' = q'~P has at least 2k vertices. Let ~ ( F , G) (resp. ~ 0 ( F , G)) denote the space of all linear transformations of F into G (resp. onto G); the spaces .~(E, G) and ~e0(E , G) are similarly defined. In the following paragraphs, it is convenient to topologize the spaces E, F, and G by means of Euclidean metrics, the spaces -~¢(F, G) and -~(E, G) by means of uniform norms, and the space ~ of all convex polygons in G by means of the Received December 3, 1962 * Research supported in part by the National Science Foundation, U.S.A. (NSF-GP-378).
1
2
VICTOR KLEE
[March
Hausdorff metric. However, at the cost of some extra effort it would be possible to dispense with these assumptions and carry out the proof for vector spaces over an arbitrary ordered field. Noting that the function ~,~P] ~ • ~e(F, G) maps £P(F, G) continuously into ~ , that the function (number of vertices of K) ] K • ~ is lower semicontinuous on g(', and that the set J = (~k•Lo(F,G):~b is biunique on ~V} is a dense open subset of ~ ( F , G), we see the possibility of choosing q in J close to ~/' such that the convex polygon Q = ~/¢P has at least 2k vertices. When Q has at least 2k + 2 vertices, there is no problem. When Q has only 2k vertices, we shall produce in G another linear image Q + of P which has more vertices than Q. Production of Q+ depends on the following simple fact: (?) I f Q is a convex polygon in the plane G, A is the set of all vertices of Q, and B is a finite subset of Q ,.~ A, then there exists e > 0 such that whenever B + is a finite set for which B + eg Q but B + lies in the e-neighborhood of B, then the polygon Q÷ = conv(A d B +) has more vertices than Q has. Here it is essential that the set B should not include any vertex of Q, and this accounts for our interest in transformations of E which are biunique on the set V of vertices of P. Suppose Q has only the 2k vertices ---ql .... , +-qk. For each i there is a unique vertex Pi of P such that r/¢pi = qi. Let H be a k-dimensional linear subspace of E such that {Pl . . . . ,Pk} C H and let ¢ denote the restriction of t/~ to H. Then of course ~(P 0 H) = Q. Let u • P ~ H, so that each point x of E admits a unique expression in the form x = x' + x"u with x' • H and x" • R (real numbers). For each point z • G, let the transformation ~b, e ~ o ( E , G) be defined as follows: 4~,(x' + x"u) = ~x' + x"z.
Note that with Zo = r/~u, we have ¢~o(X' + x"u) = ~x' + x"rl~u = nCx' + x"~lCu = rl~(x' + x"u), so ¢,o=n~. Let Go denote the set of all z e G for which the transformation ~bz is biunique on V. Then z o • Go, and we claim further that the set G ,-, G O is finite. Indeed, if z e G ~ Go there exist v • V and w • V such that v • w but ~b,v = ~b:w, whence ~(v' - w') = (w" - v")z. If v" = w", then ~v' = ~w' and r/~v = r/~w, contradicting the choice of r/. If v"=/=w", then z = ( w " - v " ) - l ~ ( v ' - w'). Thus there are only finitely many possibilities for z e G ~ Go. Let zl ~ Go ~ Q. Since Zo e Go and the set G ~ Go is finite, there exists zl/2 • Go such that the segments [z o, z~/2] and [z~/2, zl] both lie entirely in Go. Define z~ = ( 1 - 2 2 ) z o + ( 2 2 ) z l / 2
for 0 < 2 < 1 / 2 ,
zx = ( 2 - 2 2 ) z x / 2 + ( 2 2 - 1 ) z x
for 1 / 2 < 2 < 1 .
Then the function Note that
~,aPl,~ • [0,1]
and
is a continuous mapping of [0,1] into ~ .
1963]
ON A CONJECTURE OF LINDENSTRAUSS ~bzaP D ( ( H A P ) = Q
3
for all 2 ~ [0, 1], and that
~bzoP = ~/~P = Q
while
u ~ P and ¢~1u = zl(~Q.
Let p be the least upper bound of those 2 e [0,1] for which CzxP = Q. Then # < 1 and ¢z~P = Q, but there are values of 2 arbitrarily close to # for which Q is properly contained in the polygon ¢,aP. Of course CzxP = conv¢,~V for all 2 e [0, 1], and since zu e Go we have ~bz,(V "~ { + P l ..... +- Pk}) = Q "~ {+- ql ..... +- qk}Applying the italicized statement (t) above, we see the existence of 2 close t o / t for which the convex polygon Q + = tk~xQ has more vertices than Q and hence, being centrally symmetric, has at least 2k + 2 vertices. The proof is now completed by mathematical induction.| It seems worthwhile to provide a more metric form for (I). (H) I f P is an n-dimensional centrally symmetric convex polyhedron in E', there is a 2-dimensional plane T in E n such that the orthogonal projection z~ of E" onto T carries P onto a convex polygon uP having at least 2n vertices. Proof. By (I) there is a linear transformation z of E" onto /~2 such that zP has at least 2n vertices. Let T 1 = z-1(0) and let T be the orthogonal supplement of T 1 in E'. | Proof of (Z). Let P be an n-dimensional centrally symmetric convex polyhedron, centered at the origin of an n-dimensional real vector space E. Let ( , ) be an inner product on E and let pO be the polar body { y ~ E : ( x , y ) < 1 for all x ~ P}. Then of course p0 is an n-dimensional centrally symmetric convex polyhedron, and by (H) there exists a two-dimensional linear subspace T of E such that the convex polygon ztP° has at least 2n vertices, where u is the orthogonal projection of E onto T. Since TA(PNT) °=T~clconv(P°UT
O) = T N ( P
° + T O) =
T N ( riP° + TO) = ~ze°, it follows that the polygon P 0 T has at least 2n vertices. (The relevant basic results on the polarity o may be found in [1] and in [ 2 ] . ) | In the absence of symmetry assumptions, the methods used for (H) and (~) lead to similar results in which the number 2n is replaced by n + 1. In the nonsymmetric version of (E), there exist 2-dimensional sections with at least n + 1 vertices through each interior point of the n-dimensional convex polyhedron P. There remain many interesting problems concerning the numbers of faces of sections or projections of convex polyhedra. Some of these may be formulated as
4
VICTOR KLEE
follows. Suppose ~ is an indexed family ((P,, X,): t e I}, where, for each t ~ I, P, is an n-dimensional convex polyhedron in E" and X, is a nonempty subset of E". For 0 ~ j < k =<_n, let Y.(~, k , j ) denote the largest number r such that for all t e l and (*) for all x ~X,, there is a k-dimensional flat F through x for which the intersection P, f')F is a k-dimensional convex polyhedron having at least r j-dimensional faces. For various choices of ~ , and for given j < k, it would be of interest to determine the number Y(~, k , j ) and to describe the "minimal members" of ~ - - t h a t is, those members (P,,X3 of P such that (*) fails for r > Z ( ~ , k , j ) . Of special interest are the family 6~, of all pairs (P, {x)) for which P is an n-dimensional convex polyhedron which is centrally symmetric about x, and the family ~ , of all pairs (P, int P) where P is an n-dimensional convex polyhedron. I am indebted to M. Perles for some helpful comments. REFERENCES 1. Bourbaki, N., 1955, Espaces veetoriels topologiques, Chaps. 3-5, A.S.I., 1229, Hermann, Paris. 2. Klee, V., 1959, Some characterizations of convex polyhedra, Acta. Math., 102, 79-107. 3. Klee, V., 1960, Polyhedral sections of convex bodies, Acta. Math., 103, 243-267. UNIVERSITY OF WASHINGTON SEATTLE, WASH|NGTON, U.S.A.
STRICTLY ANTIPODAL SETS* BY
BRANKO GRiJNBAUM ABSTRACT
A subset A of E3 is called strictly antipodal provided that for every pair Xt, X2 of points of A there is a pair//1, //2 of parallel supporting planes of A such that H~t'3 A = (X~}. The main result asserts that a strictly antipodal set has at most five points. This strengthens a recent result of Croft [2]. 1. Introtluetion. For a convex polyhedron K let v(K) denote the number of vertices o f K. I f K1 and K 2 are convex polyhedra it is clear that v(Kl + K 2 ) < v(K1).v(K2). It is easy to find examples showing that equality may hold for suitable K 1 , K 2 in E3; i f K I , K 2 , c E 2, then
v(KI + K2) <=v(KO + v(K2). More complicated, and unsolved in the general case, is the following related problem: I f K is a convex polyhedron in E", with v -- v(K) vertices, how m a n y vertices can K * = K + ( - K ) have? It is easily checked that, independent of n, v(K*) <=v(v - 1). However, equality in this relation can take place only if v is not too large with respect to n. Let f ( n ) denote the maximal v such that there exists an n-dimensional convex polyhedron K with v = v(K) and v(K*) = v(v -- 1). It is easily seen that f ( 2 ) = 3. In Section 2 we shall prove the following result.
THEOREM. f ( 3 ) = 5. As easy corollaries we shall obtain (in Section 3) a simple solution of a problem of Erd6s 1-5] recently solved by Croft [2], as well as a number of results on families of translates of convex polyhedra in E a. Some additional remarks and problems are also given in Section 3. 2. Proof of the Theorem. For an arbitrary set A c E" let a pair of points X a , X 2 c A be called strictly antipodal if there exists a pair H1,H 2 of (distinct) parallel supporting hyperplanes of A such that A n Hi = {Xi} for i = 1,2. A set A is called strictly antipodal provided ever y two points of A are strictly Received December 19, 1962 * This research was supported in part by the United States Air Force under Grant No. AF-EOAR 63-63 and monitored by the European Office, Office of Aerospace Research. 5
6
BRANKO GRUNBAUM
[March
antipodal. Let g(n) denote the maximal number of points in a strictly antipodal set A c E". The following assertion is obvious: LEMMA. The set of vertices of a convex polyhedron K is strictly antipodal if and only if v(K*) = v(K) .(v(K) - 1). If A is a strictly antipodal set and K its convex hull, then A coincides with the set of vertices of K. Therefore, the lemma implies that f ( n ) = g(n). In order to prove the Theorem it is sufficient to show that the common value o f f ( 3 ) and of g(3)is 5. Since f ( 3 ) > 5 (see Section 3) and since every subset of a strictly antipodal set is strictly antipodal, we have only to show that no 6-pointed set in E a is strictly antipodal. Assuming this to be false, let A be a six-pointed strictly antipodal set in E 3, with convex hull K. Because o f f ( 2 ) = g(2) = 3, all the faces of K are triangles. Counting incidences and using Euler's formula it follows at once that K must be a polyhedron of one of the two types represented in Figure 1 by their Schlegel diagrams.
D
B
Figure 1 We first note that K cannot have configuration II. Indeed, if there would exist such K, in view of the affine invariance of strict antipodality we could assume that K has the form indicated in Figure 2. Since the segment E F is not an edge of K, min {a,~} < 1. Without loss of generality we assume that a < 1, and we note
cf
A
E
Figure 2 A = (1,0,1); B = ( 1 , 0 , - 1 ) ; C = ( - 1 , 1 , 0 ) ; D = (-1,-1,0);
E = (a,b,c); F = (o:,fl, y).
1963 ]
S T R I C T L Y A N T I P O D A L SETS
7
that E is contained in the wedge formed by the planes through ACD and BCD. Considering the sections of K by the planes z = 0 resp. z = c it is immediate that C and E are not strictly antipodal. Thus we may assume, for the remaining part of the proof, that K has configuration I; without loss of generality we may therefore assume that K has the form indicated in Figure 3. ¢
F
E
g~
D Figure 3
A =
(-1,-1,0);
8 = (-1,1,0);
C = (1,0,1);
O = ( 1 , 0 , - 1 ) ; E = (a,b,c); F = (a, fi, T). Since sufficiently small displacements of the points of a strictly antipodal set do not destroy strict antipodality, it follows that no generality is lost in assuming that no edge of K i s parallel to a face o f K , and that I b[ + [c[ ~ 1, [fll + [Vl ¢ I. It follows that K* = K + ( - K) will have 16 triangular faces (called t-faces in the sequel), and that all other faces of K* are parallelograms (called p-faces in the sequel). Using again a count of incidences and Euler's formula, it is easily found that K* has 20 p-faces. We shall end the proof of the Theorem by examining the construction of K* and by showing that it cannot contain 20 p-faces. Let K 1 denote the convex hull of {A, B, C, D, E}. Then K* is a polyhedron with the vertices + _ ( C - A ) = + ( 2 , 1 , 1 ) , _(D-A)=+(2,1,-1), +_(C-B)= -I-(2,-1,1), + _ _ ( D - B ) = _ _ _ ( 2 , - 1 , - 1 ) , ___(B-A)= +__(0,2,0), + ( C - D ) = +(0,0,2), _+ (E - A) = ___(I + a, 1 + b,c), + ( E - B ) = _ + ( 1 + a,-l+b,c), +__(E- C) = +_(-1 + a , b , - 1 + c), +_(E- D) = +__(-1 + a,b, 1 + c). Since K is of type I, we have a > 1; this and the convexity of K* imply that I bl + I c I < 1. (Indeed, by considering the vertices C - A, D - A, C - B, D - B, E - A and E - B it follows from a > 1 that I c[ < 1. Then, assuming without loss of generality that c > 0 and b + c > 1, a consideration of the vertices C - D, C - B, E - B, E - D, leads to the contradiction that C - A is not a vertex of K*). Projecting orthogonally the part of K~ contained in the half-space E + = {(x, y, z)l x > 0} onto x = 0, we obtain a configuration of the type represented in Figure 4a. Denoting by K 2 the convex hull of {A, B, C, D, F}, the same reasoning applied to K~ leads to a configuration of the type given in Figure 4b.
8
BRANKO GR(INBAUM
Figure 4a
[March
Figure 4b
Now, K* is the convex hull o f K ~ u K * u {E - F , F - E}. Obviously, E - F and F - E are incident only to t-faces of K*; therefore the number of p-faces of K* at most equals the number of p-faces in the convex hull Q of K* t3 K*. But the latter number is at most 12. Indeed, superimposing Figures 4a and 4b, we observe that every p-face of Q is a p-face of either K* or K*, and that only one p-face of Q contained in the half-space E* can be incident to each of the vertices C-A,D-A,C-B,D-B.
Thus Q has at most four p-faces contained in E ÷ ; by symmetry the same number of p-faces of Q is contained in the half-space E. Together with the four p-faces parallel to the x-axis, this yields at most 12 p-faces for Q, and thus also for K*, in contradiction to the former assertion that K* has 20 p-faces. This completes the proof of the Theorem.
3. Some related results and problems. i) Erd~Ss [5 3 posed the problem of determining the maximal number e(n) of points in E ~ such that all the angles determined by triplets of the points be acute. For n = 3 Croft [2] recently established that e(3) = 5. This results also from our Theorem and from the obvious assertion e(n) < g(n). ii) The inequality e(n) >= 2 n - 1 was established in [3] by means of the following example (reproduced here for the sake of completeness): Let {e ~}n~= 1 be mutually orthogonal unit vectors in E ". The (2n - D-pointed set { A , B a , ' " , B , , C2, " " , C , satisfies Erd6s' condition if, e . g . , A = ea, Bk = ~kel + ek, Ck = -- eke1 -- % k = 2, 3,..., n, where all ~k'S satisfy 0 < ~k < 1 and are diflerent from each other. iii) As mentioned (in part) in [3], it is easily shown that g(n) is also the maximal number of members in any family ~ of translates of a convex body K c E", provided the family satisfies any of the following conditions: (a) The intersection of any two member.s of oF is a single point; (b) The intersection of all members of s f is a single point, which is also the only common point of any two members of S ; (c) The intersection of any two members of o~/"is (n - 1)-dimensional. The same is true if in (a) or in (c) the attention is restricted to centrally symmetric K.
1963]
STRICTLY ANTIPODAL SETS
9
iv) The restriction o f ~ to families of translates of one convex body is essential in iii). This is obvious in case of conditions (a) and (b); in the case of (c) the bound is 4 for n = 2 (while g(2) = 3); already for n = 3 it has been proved repeatedly (e.g. by Tietze, Besicovitch, Rado; see [4] for references to these and some related results) that there exists no finite bound. In [4] it is also pointed out that arbitrarily large families X in E 3, any two of whose members have a 2-dimensional interzection, are obtainable as the Schlegel-diagrams of the duals of g-dimensional "neighborly polytopes" (see [7]). This was known, however, already to BrOckner [1]. Nevertheless, the following question seems to be open even for n = 3: How many members can a family of centrally symmetric convex bodies in E" have, if every two have an (n - 1)-dimensional intersection? v) Among unsolved problems related to the Theorem of the present paper we mention: (a) The determination of e(n) and of J ( n ) = g(n) for n > 3; in particular, is e(n) = g(n) for all n? (b) The determination of h(k,n) = max {v(K + ( - K ) ) I K c E n, v(K) = k}
for k > 2n, n > 3. REMARK. The example ii) above implies h(k, n) = k ( k - 1) for n + 1 < k < 2n - 1 ; for k > n = 2, we have h ( k , 2 ) = 2k. This follows from the observation that h(k, n) = 2s(k, n), where s(k, n) is the maximal number of strictly antipodal pairs of vertices for a convex polyhedron K c E" with v(K) = k, and the assertion s(k,2) = k. To prove this latter assertion assume that s(k,2) > k for some k. Let ko be the minimal k with this property and let K, with v(K)= ko, be a ko-gon such that more than k o pairs of vertices of K are strictly antipodal. Then at least one vertex Vo is antipodal to some three consecutive vertices Vi-1, Vu V~+1 of K; but then V~is easily seen to be strictly antipodal only to Vo. Thus the convex hull of the k o - 1 vertices of K different from V~ yields an example showing s(ko - 1, 2) > ko - 1, in contradiction to the minimality of k o. It is worth mentioning that s(k,3)>__ [ k / 2 ] . [ k ( + 1)/2] + 2 for k >__4, the difference in behavior between s(k, 2) and s(k, 3) being similar to the jump in the number of times the diameter of a set is assumed in 3- and 4-dimensional sets (Erd~Ss [6]). The above inequality is easily established by placing approximately half of the points on each of two suitable circular arcs. vi) Klee [8] defined a pair of points X1, X 2 of a set A c E" to be antipodal provided there exist distinct parallel supporting hyperplanes HI, H 2 of A such that Xi e A c3 Hi, i -- 1, 2; he also asked about the maximal number of points in a set A c E" such that every two points of A are antipodal. It was established in [3] that the required number is 2". In analogy to the above definition of s(k, n) one may ask what is the maximal number a(k,n) of pairs of antipodal points in k-pointed sets in E". While the problem is open for n > 3, it can be shown by
10
BRANKO GRLINBAUM
arguments similar to those a(k, 2) = [3k/23.
used above in connection
with
s(k, 2) that
REFERENCES
1. Briickner, M., 1909, Uber die Ableitung der allgemeinen Polytope und die nach Isomorphismus verschiedenen Typen der allgemeinen Achtzelle (Oktatope), Verb. NederL Akad. Wetensch. Sect. I, 10 (1), 27pp. + 2 plates. 2. Croft, H.T., 1961, On 6-point configurations in 3-space, d. London Math. Soc., 36, 289-306. 3. Danzer, L. and Grfinbaum, B., 1962, lJber zwei Probleme beziiglich konvexer K~rper von P. ErdiSs und von V.L. Klee, Math. Z., 79, 95-99. 4. Danzer, L., Griinbaum, B. and Klee, V., Helly's theorem and its relatives. Proe. Symp. Pure Math., Vol. 7, 101-180. 5. Erd~s, P., 1957, Some unsolved problems, Michigan Math. J., 4, 291-300. 6. ErdiSs, P., 1960, On sets of distances of n points in Euclidean space, Magyar Tud. Akad. Mat. Kutat6 lnt. K6zL, 5, 165-169. 7. Griinbaum, B. and Motzkin, T.S., On polyhedral graphs, Proc. Symp. Pure Math., Vol. 7, 285-290. 8. Klee, V., t960, Unsolved problems in intuitive geometry, (Mimeographed notes)Seattle. THE HEBREW UNIVERSITY OF JERUSALEM
Added in p r o o f (June 13, 1963): The result of Croft [2] that e(3) = 5 was recently established also by Schiitte, K., 1963, Minimale Durchmesser endlicher Punktmengen mit vorgeschriebenem Mindestabstand, Math. Annalen, 150, 91-98.
BOUNDS FOR THE PRINCIPAL FREQUENCY OF NONUNIFORMLY LOADED STRINGS* BY
BINYAMIN SCHWARZ ABSTRACT
n particles (beads) of different masses are mounted at equally spaced points along a string which in itself is weightless, of unit tension and fixed at its endpoints. Given the set of masses, it is shown that the principal frequency of this loaded string becomes minimal if they are arranged decreasing from the center in as nearly a symmetrical order as possible. If a strictly symmetrical increasing arrangement of the masses exists, then this arrangement gives the maximum of the principal frequency. 1. Introduction. In a paper by P. R. Beesack and the author [1] differential systems corresponding to strings with continuous, but nonhomogeneous, density p(x) were considered. Setting the constant tension of the string equal to 1 and fixing it at its endpoints x --- 0 and x = L, the square of the principal frequency is the least characteristic value 21 = 2x(p) of (1.1)
y"(x) + 2p(x)y(x)=O, y(O)= y(L)=O;
(p(x)>O, O< x< L).
Together with the given density p(x) all densities equimeasurable to it in [0, L] were considered. It was proved that among all strings of such a class the least principal frequency occurs in the case that the density is symmetrically decreasing about the midpoint o f the string and the maximum principal frequency occurs when the density is symmetrically increasing. These extremizing densities, the symmetrically decreasing and increasing rearrangement of p(x), were denoted by p-(x) and p+(x) respectively. The result just described was hence formulated as (1.2)
21(P-) < 2t(p) <_-2t(p+),
[1, Theorem 2] and it is easily seen that the corresponding equality holds only if p = p - or p = p+, respectively [See 8, Theorem 1]. In the present paper we consider the analogous question for the discrete case. n particles (beads) of different masses are mounted at equally spaced points along a string,which in itself is weightless and of unit tension. The string is again fixed at Received January 17, 1963 (Revised April 28, 1963) * Sponsored by the Mathematics Research Center, United States Army under Contract No. DA-11--022-ORD-2059, University of Wisconsin, Madison. 11
12
BINYAMIN SCHWARZ
[March
its endpoints x = 0 and x - L and the mass rni is attached at x i = i L / ( n + 1), i = 1.... , n. Let y~ be the amplitude of the ith particle for a harmonic oscillation of this loaded string. The column vector y = (Yl .... , y,) satisfies (1.3)
A2y = 2Py,
where A2 = A2~")isthe Jacobi (tri-diagonal) matrix of order n: f
(1.4)
2 -1
-1 2
-1
A~2") = A2 = -1
2 -1
-1 2
2 is the square of the frequency, and (1.5)
P = {Pl ..... P,}
is the diagonal matrix of order n whose elements are p~ = m i L / ( n + 1) i = 1,..., n I-4, Chapter III, and 6, Chapter 3]. (As in future only the pi's and not the mi's appear, we shall refer to Pi as the mass of the ith bead). The square of the principal frequency is the least characteristic value 21 = 21(P ) of the pencil A - 2P; i.e. 21(P) is the smallest root of the characteristic equation ]A - 2P] = 0. Our problem is the following: Given a set of beads, how are they to be arranged at the equidistant points x~ in order to extremize 21 ? In contradistinction to the continuous case, there exists in general no arrangement which is strictly symmetrical with regard to the midpoint x = L / 2 of the string. Nevertheless, the analogue o f 2 1 ( p - ) < 21(p ) holds in the following sense: if the beads are arranged in as nearly symmetrically decreasing order as possible, with the inevitable overweight given at each step to one and the same side, then 21 takes its least possible value.This is essentially Theorem 1 o f §3; however, we formulate this result in a slightly more general way by replacing A 2 by A k (see (2.6)). This theorem is the main result of our paper. §2 contains the necessary definitions and two lemmas. Lemma 2 is of special importance for the proof of Theorem 1. Its first part follows from a theorem of Hardy, Littlewood and P61ya 1-5, Theorem 371] ; its second part, needed for a uniqueness statement in Theorem 1, seems to be new. Both parts of this lemma follow from a result on rearrangements proved by A. Lehman [7]. In §4 an inequality corresponding to 21(p) < 21(p +) is established but only for the case where a stricly symmetrically increasing arrangement of the beads exists.
1963]
BOUNDS FOR FREQUENCIES
13
2. Definitions, notation and two lemmas. We start with the following definitions which are slightly different from those used in the book of Hardy, Littlewood, and P61ya [5, Chapter X]. An ordered set ( a ) = (al .... ,an) of n real numbers is called symmetrically decreasing if either (2.1)
al < an < a2 ~ an-1 < ... < a[(n+2)/2]
o1"
(2.2)
an < al <
an-1
~ a 2 <= . . .
~ at(n+l)/21
holds. The set (a) is symmetrically increasing if either (2.3)
at > an > a2 > an-1 > ... > a[(n+2)/21
or
(2.4)
an ~ a l > a n _ 1 ~ a 2 ~ . . . ~
a[(n+l)/2]
holds. The set (a) is strictly symmetrically increasing if (2.5)
al = an >=a 2
=
an-1 > ... > (=)a[(n+ 2)/z].
For a given set (p) = (Pl ..... Pn) there exist, in general, two distinct, symmetrically decreasing rearrangements. The rearrangement ordered as in (2.1) is denoted by ( p - ) = (p;-, ...,p~) so that (2.1')
Pl- < P~ < P ; < P~--1 < ... --< Pt~n+2)/21 ;
the other symmetrically decreasing rearrangement is denoted by ( - P ) ----( - P l , . . . , - P,) :
(2.2')
-Pn =< -Pl --<- Pn-1 --< -P2 --< ... --< -Pt(n+t)/2~.
Similarly, the symmetrically increasing rearrangements of (p) are denoted by (p+) and (+p) and their elements satisfy (2.3')
P
> P
> P =
> P =
1=
... = P[(n+2)/2]
and (2.4')
+ pn => + p1>=+ p n--1 => + p2>-~
> +P[(n+l)/2]"
"'" =
Finally, if (p) admits a strictly symmetrically increasing rearrangement, i.e. if (p +) = (+p), then this rearrangement is denoted by (p*) = (p*, ..., p*): (2.5')
=
-----
= P n* - t
>---- "'" ~ ( --- ) P [ ( n* + 2 ) / 2 ] "
Sets (p) admitting such a rearrangement (p*) are called paired. For even n a set (p) is paired if every value occurs an even number of times; for odd n the smallest
14
BINYAMIN SCHWARZ
[March
value has to occur an odd number of times, every other value an even number of times. If (q) = (ql ..... q,) is a rearrangement of (p) = (Pl ..... p,) then we say that the diagonal matrix Q = (ql ..... q,} is a rearrangement of the diagonal matrix P = {Pl, ..., P,}. Given P, its rearrangements P - , - P , P+, +P and, if P is paired, P* are defined by the order relations (2.1')-(2.5') for their elements. Together with these diagonal matrices of order n, we consider also Jacobi matrices of the same order for which the elements in the diagional are equal to the real constant k and the elements in the super- and sub-diagonal are - 1. (All other elements are 0). We use the notation (cf. (1.4)) k -1 (2.6)
-1 k
-1
A~" ) = Ak = -1
(Aktx) = ( k ) , A~k2 ) =
k -1
-1 k
..). For the corresponding quadratic forms
1
the following two lemmas hold: LEMMA 1.
Let n--I
(2.7)
A~k")(y,y) = Ak(y,y ) = k ~ y2 _ 2 ~ YiY~+I /=I
i=i
be the quadratic form belonging to the matrix (2.6). (i) I f k > 2cos(Tr / n + 1) then Ak(y , y) is positive definite; (ii) if k < 2 cos (it / n + 1) then Ak(y , y) takes negative values; (iii) if k = 2cos(~r/n + 1) then Ak(y,y ) is non-negative definite. This lemma is an immediate consequence of the following theorem due to Fan, Taussky and Todd [2, Theorem 9]. I f yl,..., y, are n real numbers, then l1
•
i=0
2
7~
]~
2
( Y , - Yi+ 1)2 > 4sin -~n-+-])',=o y'
(where Yo = Y.+x = O) unless Yi = c~, where (2.8)
Yi= s i n - - - - n+l'
i = 1,.
""
n.
LEMMA 2. Let Ak(y, y ) be defined by (2.7) and let the set (y) of n non-neoative numbers be given except in arrangement. Then Ak(y , y) attains its minimum if
1963]
BOUNDS FOR FREQUENCIES
15
(y) is arranged in symmetrically decreasing order. Moreover, if all the elements of (y) are positive and if no three elements of (y) have the same value, then Ak(Y, Y) attains its minimum only if(y) is symmetrically decreasing. In our notation, this can be stated as (2.9)
Ak(y, y) > Ak(y-, y- ),
y, > O, i = 1,..., n,
with the additional statement that, if Yi > 0, i = 1, ..., n and if no three numbers Yi are equal, then equality holds in (2.9) only if (y) = ( y - ) or (y) = (- y). We remark that ~i=xYi n 2 is invariant for all rearrangements of a given set. Defining
S(y,y) : ~ y2+ ~ YiY,+I,
(2.10)
i=1
i=I
it follows that (2.9) is equivalent to
S(y,y) < S ( y - , y - ) .
(2.11)
This last inequality is a very special ease of a Theorem of Hardy, Littlewood and P61ya on bilinear forms. [5, Theorem 371; to obtain (2.11) set, in their notation, c o = 1, c 1 = c_ 1 = ½, all other c = 0; let their two sets (x) and (y) coincide and if (our) n is even, let one element of (y) be zero.] This proves the first part of Lemma 2. As mentioned, both parts of the lemma follow from Lehman's result. Indeed, if we apply the corollary of [7] to the funetionf(x) = x 2 we obtain that n-1
~ YiYi+l,
i=1
i=l,..'n,
yi>=O,
is maximum if (y) is symmetrically decreasing and that under the more restrictive assumptions on (y) this maximum is attained only if (y) is symmetrically decreasing. This is clearly equivalent to Lemma 2.
3. The minimum of the least characteristic value. THEOREM 1. Let Ark")= A k be the Jacobi matrix of order n defined by (2.6) and let Q = {ql .... ,q,), qi > O, i = 1. . . . ,n be any diagonal matrix of order n with positive elements. Denote the least characteristic value of the pencil A k - 2Q by 2t(Q). Let P = {Pl .... ,P,}, P i > O, i = 1.... ,n be a given diagonal matrix and let P - and P + be its symmetrically decreasing and increasing rearrangement respectively.
if (i)
k > 2 cos
7[
n '+ 1'
16
B I N Y A M I N SCHWARZ
[March
then (3.1)
2x(P) _>_2~(P-).
Moreover, if 7~
(i')
2 > k > 2cos----~, =
n-t-
then equality holds in (3.1) only if P itself is symmetrically decreasing.
if 7g
(ii)
k < 2 cos - - - - n+l'
then (3.2)
2~(P) > 21(P + ).
Finally, if 7~
(iii)
k = 2 cos ~ n+l'
then (3.3)
2a(Q) = 0
for any diagonal matrix Q = {ql . . . . . qn}, qi > O, i = 1..... n. Proof. For given n and k let y = (Yl ..... Yn) be the characteristic (principal) column vector corresponding to the least characteristic value 2~(P) of the pencil A k - - 2P[3, p. 310]. Let lyl = (I yt [ .... ,1 y,[), then n-1 i=l
[2
n
Ak(y,y)= k ~ y ~ - 2 ]E y i Y i + l i=l
>=k
Ely /
n-1
-2]E
i=l
i=1
lyillyi+xl= Ak(lyl, lyl),
and
P(y,y)= ~ piy~= ~ i=l
i=l
p, Iy, I
=P(lYI,lYl).
It now follows from the minimum characterization of 2x(P) [3, p. 319] that [y[ is also a characteristic vector corresponding to 2t(P). But the recursive form of the n scalar equations of (3.4)
Aky = 21(P)Py
shows that the characteristic vector belonging to 21(P) is determined to within a scalar factor. We may therefore assume y = and it follows from (3.4) that y > 0, i.e. y~ > 0 for all i, i = 1..... n. ['Cf. 4, p. 136].
l yl
1963]
17
BOUNDS FOR FREQUENCIES
Assume now that (i) holds. By Lemma 1, Ak(y, y) is in this case positive definite, hence 21(P) > 0. Let y be the positive characteristic vector belonging to 21(P) Then
(3.5)
Ak(x,x) = 21(p ) = Ak(y,y__)_ > A k ( Y - , Y - ) • > min P-(x,x'-----~) P(Y, Y) = P - ( Y - , Y - ) = :,,o
0).
Here y - = (YT, ..., Y~) is the symmetrically decreasing rearrangement of y: (3.6)
(0 <)y~- < y~ < Y2 < Y~-I < ... < Y[(n+2)/2]"
(3.6) and (2.1') show that y- and P - are similarly ordered and it follows by a well-known theorem of Hardy, Litttewood and P61ya [5, Theorem 368] that (3.7)
p ( y , y ) = ~ piy2 < ~ p.i-y.[-2 = p - ( y - , y - ) . i=1
/=1
(3.7) and the first part of Lemma 2 (i.e. (2.9)) imply the first inequality sign of (3.5). The minimum in the fourth term of (3.5) is taken over the class of all not identically vanishing vectors x, and y - ( > 0) clearly belongs to this class. By the minimum characterization of the least characteristic value, this minimum is ~l(P-). This proves (3.5) and hence also (3.1). For the next assertion of the theorem we have to apply the second part of Lemma 2. We already know that, for any k, the characteristic vector y corresponding to 2t(P) can be chosen in such a way that all its components y,, i = 1.... , n are positive. It remains to be shown that, if (i') holds, then no three components Yi have the same value. But in this case (3.4)implies (as 2~(P)> 0, Pi > O, Yi > O, i = 1, ...,n) (3.8)
Aty > 0
or, explicitly, (3.9)
kyi>yi_l+yi+l,
yt>0,
i=l,...,n,
(Yo=Yn+x=0).
(3.9) and (i') give (3.10)
2Yi > Yi-1 + Yi+l,
y~> O, i = 1,...,n,
(Yo = Y,,+I = 0).
i.e. y > 0 is strictly concave. This strict concavity implies that no three components y~ can have the same value and y thus fulfills the conditions required for the second part of Lemma 2. Let k now be restricted to the range given by (i') and assume that equality holdsin (3.1), i.e. (3.11)
2x(P)
=
2i(P-).
18
BINYAMIN SCHWARZ
[March
The inequality signs in (3.5) thus become equality signs. As the minimum in the fourth term of (3.5) is obtained only for the characteristic vector of Ak -- 2P- it follows that y - is this vector: (3.12)
Aky- = 2t(P-)P y .
Equality of the first two ratios of (3.5)implies (by (3.7) and the first part of Lemma 2, i.e. (2.9)) Ak(y, y) = At,(y- , y - ) . It thus follows from the second part of Lemma 2 that either y=y-
(3.13) or
y ~
(3.14)
--y.
Here - y = (-y~,...,-y,) is the other symmetrically decreasing rearrangement of y, satisfying (3.15)
(0 < ) -y,, < - Y l < -Y,,-x < -Y2 < ... <=-Yan+x)/2l.
(3.4), (3.11), (3.12) (3.13) imply P = P - . (2.1'),(2.2') and (3.6),(3.15) imply that -Pi = P,-+1-i and -Yi = Y ~ l - i , i = 1, ...,n. (3.12) gives thus (3.16)
A k - y = 2x(P-) -P -Y.
(3.4), (3.11), (3.14) and (3.16) imply P = -P. Hence, if (i') holds, equality in (3.1) implies that P is symmetrically decreasing. The proof of (3.2) is analogous to the proof of (3.1). If (ii) holds, then (by Lemma 1) Ak(y, y) takes negative values and 21(P ) is thus negative. (3.5) is thus replaced by (3.17)
• Ak(X,X) (0 >)21(P)= Ak(Y'Y) > A k ( y - , y - ) > mln o--¥~,---= 21(p+), P(Y,Y) = P + ( Y - , Y - ) = ~,~o P (x,x)
which follows by (2.9) and n
(3.18)
p ( y , y ) = ]Elp,y ~ >= ~, p+ y ; 2 = p + ( y - , y - ) . i =~
5=1
(3.18) follows by I'5, Theorem 368] as y - and P+ are oppositely ordered. Finally, if (iii) holds, ~ defined by (2.8), is clearly the characteristic vector corresponding to 21(Q) = 0 for any Q = {qa, --., q,}, qi > 0. This completes the proof of the theorem. For k = 2 we gave the mechanical interpretation of this theorem in the introduction. For k > 2 the n beads should be connected by springs of equal strength (proportional to k - 2) to the line y = 0 [-See 4, p. 269 for the analogous continuous case]. In the critical case k = 2cos(n/n + 1) the "negative" springs
1963]
BOUNDS FOR FREQUENCIES
19
and the tension of the string result in a static equilibrium displacement whose shape (Yi = c sin (in / n + 1)) is independent of the masses. 4. The m a x i m u m of the least characteristic value for k = 2 and paired P. W e now restrict ourselves to the case k = 2. It follows that the least characteristic value 21(Q) of A 2 - 2Q, ( Q = {ql .... ,q,}, qt > 0 , i = 1,...,n) is positive. For the continuous string the proof of 21(p +) > 21(p) was simpler than the proof of 21(p) > 21(p-). For n = 3 (and k = 2) an easy computation shows that 21(P +) > 21(P). But already for n = 4 this inequality is, in general, not valid. Indeed, for P = {Pt, P2, P3, P4} we obtain,
~(~,) = [A (4) _ ;~P[ = 5 - 2~(2pl + 3p 2 + 3p3 + 2p4) (4.1)
+ 22[3(piP2 + PIP4 + P3P4) + 4(piP3 + P2P3 + P2P4)] - 223(PlP2P3 + PlP2P4 + PlPaP~ + P2P3P4) + 24PlP2PaP4.
Note that only the coefficients of 2 and 22 depend on the arrangement of P. Denote the four masses by a, b, c and d and assume that (4.2)
0 < a < b < c < d.
Let (4.3)
P = {d,a,b,c}.
Its symmetrically increasing rearrangement P+ is then given by (4.4)
P+ = {d, b, a,c}.
Computing (4.1) for P and P ÷ and subtracting we obtain (using an obvious notation) (4.5)
q~(2)- ~b+(2)= (b - a ) ( d - c)22.
(4.2) and (4.5) imply 21(P) > ;tl(P+). Comparing ~b(2) of P (given by (4.3)) with the characteristic functions of all its rearrangements it can be shown that the maximum 2t corresponds to (4.3). But we did not succeed in finding the maximizing arrangement for n > 4 and general P. Indeed, it may well be that no such general maximizing arrangement exists and that the order relations defining the maximizing arrangement vary with the set of masses. However, for paired sets of masses the analogue of 21(p +) > 21(p) is valid. THEOREM 2. Let A~zn) = A 2 be the Jacobi matrix of order n defined by (1.4). Let the diagonal matrix P = {Pl .... ,pn}, p i > 0 , i = 1,...,n be paired and
20
BINYAMIN
SCHWARZ
[March
let P* = {p* ..... p*} be its strictly symmetrically increasing rearrangement. Denote the least characteristic value of A 2 - A P and A 2 - 2 P * by 2~(P) and 21(P* ) respectively. Then
(4.6)
2t(P*) > 2~(P)
and equality holds only if P = P*.
Proof. Let y > 0 be the characteristic vector of A2 - ,~P* belonging to 2~(P*). Hence, (4.7)
A2y = 2x(P*)P*y.
By (2.5) P* is "symmetric" in the sense that (4.8)
p * = P n*+ l - i ,
i = 1,...,n.
The characteristic vector y is determined to within a scalar factor and does not change signs. (4.7) and (4.8) imply therefore that (4.9)
Yi = Y,+l-i,
i = 1,...,n.
This symmetry of y and its previous established concavity (3.10) imply that y is strictly symmetrically decreasing, i.e. (4.10)
(0 <)Yl = Y. < Y2 = Y.-1 < ... Y[(n+2)/2].
(4.10) and (2.5') show that y and P* are oppositely ordered. Theorem 368 of [5] gives (4.11)
P*(Y,Y) = ~ P'Y? < ~ PiY~ = P(Y,Y). i=1
/=X
Using (4.11) we obtain (4.12)
minA2(x,x) = ;q(p)
21(P*) = A2(Y,Y____~)> A2(y,Y) > P*(y,y) = e ( y , y ) - x~o e ( x , x )
( > 0).
This proves (4.6). If (4.13)
21(P*) = 21(P),
then the inequality signs in (4.12) become equality signs. As the minimum in the fourth term is obtained only for the characteristic vector of A2 - 2P it follows that y is this vector: (4.14)
A2y = 2x(P)Py.
(4.7), (4.13) and (4.14) give P = P*. This completes the proof of the theorem. We remark that Theorem 2 remains clearly correct if A~") is replaced by A~") where k satisfies condition (i') of Theorem 1.
1963]
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21
REFERENCES 1. Beesack, P. R. and Schwarz, B., 1956, On the zeros of the solutions of second-order linear differential equations, Can. J. of Math., 8, 504-515. 2. Fan, K., Taussky, O. and Todd, J., 1955, Discrete analogs of inequalities of Wirtinger Monatsh. Math., 59, 73-90. 3. Gantmacher, F. R., 1959, The theory of matrices, Vol. I., Chelsea, New York. 4. Gantmacher, F. R. and Krein, M. G., 1960, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Akademie Verlag, Berlin. 5. Hardy, G. H., Littlewood, J. E. and P61ya G., 1952, Inequalities (2nd edition), Cambridge. 6. Langer, R. E., 1947, Fourier's series, Amer. Math. Monthly, 54, Supplement pp. 1-86. 7. Lehman, A., 1963, A result on rearrangements, Israel Jour. Math, 1, 22-28. 8. Schwarz, B., 1961, On the extrema of the frequencies of nonhomogeneous strings with equimeasurable density, 1961, J. Math. Mech., 10, 401-422. MATHEMATICSRESEARCHCENTER,U. S. ARMY,MADISON,WISCONSIN AND TECHNION-ISRAELINSTITUTEOF TECHNOLOGY,HAIFA
A RESULT ON REARRANGEMENTS* BY
ALFRED LEHMAN ABSTRACT
Circular symmetry is defined for ordered sets of n real numbers: (Y)=(Yl,"',Yn). Let f(x) be non-decreasing and convex for x ~ 0 and let (y) be given except in arrangement. Then l~= lf([ Yi-Yt + 1 ]) (where Yn+ 1 ~ Y l ) is minimal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. The problem considered here arose in connection with a paper by B. Schwarz [2]. For the sake of completeness we start with a definition given there which differs slightly from the definition given in the book of Hardy, Littlewood and P61ya [1, Chapter X]. An ordered set ( a ) = ( a l , " ' , a n ) of n real numbers is called symmetrically decreasing if either (1)
a l <=a n < a= 2
<
=
an-I
< = ...<
<
a[(n+2)/2]
or
(2)
a, < a 1 < an_ 1 <=a 2 ~ ... <=attn+l)/z ]
holds. For a given set (y) = (Yl, "",Y,,) there exist, in general, two distinct symmetrically decreasing rearrangements. The rearrangement ordered as in (1) is denoted by ( y - ) = (y~-, .--, y~ ) so that (1')
Y-I <=Y~ <=Y'£ <=Y~-I <= "'" <=Y-~,+2)/z];
The other symmetrically decreasing rearrangement is denoted by
(-y) = (-y....,-y~): (2')
-Yn < -Yl =< -Yn-1 =< -Y2 -<--"'" =< -Yt(,+I)/21.
We add the following definitions. A circular rearrangement of an ordered set (Y) = (Y~, " ' , Yn) is a cyclic rearrangement of (y) or a cyclic rearrangement followed by inversion. For example, the circular rearrangements of the set (1,2, 3,4) are the sets Received January 17, 1963 (Revised April 28, 1963). * Sponsored by the Mathematics Research Center, United States Army under Contract No. DA-11-022-ORD-2059, University of Wisconsin, Madison. 22
A RESULT ON R E A R R A N G E M E N T S
(1,2,3,4),
(2,3,4;1),
(3,4,1,2),
(4,1,2,3),
(4,3,2,1),
(1,4,3,2),
(2,1,4,3),
(3,2,1,4).
23
An ordered set (y) = (Yl, "", Y,,) of n real numbers is arranged in circular symmetrical order or is of circular symmetry if one of its circular rearrangements is symmetrically decreasing. It follows that the sets (y -), ( - y ) and (a), satisfying (1) or (2), are of circular symmetry, and so is the set (b) = ( b l , ' " , b~) if either (3)
bl <=b2 <- bn < b3 <=bn-1 < ""< b[(n+3)/2]
or
(4)
b2 < bl < ba <=bn < b4 < b,,-1 < "'" < b[(n+4~/2]
holds. Sets of circular symmetry with a given set of n elements can be visualized as follows. Let c i, i = 1,-.., n be the n given numbers and assume that c 1 < c2 <.." < c,. Place these n numbers at n distinct points of a circle according to the following rule: cl may be put at any point; c 2 is a neighbor of c~ ; c 3 is the other neighbor of c~ ; c~ is a neighbor of c2 ; c5 is a neighbor of ca, etc. (We could also start with the largest element c,,.) The figure shows the construction for n = 8. The sets of circular symmetry are obtained by starting at any point and going in clockwise or counter-clockwise direction over the circle. C8
C4
Ct
Figure We state now our result. THEOREM. Let the ,function f ( x ) be non-decreasing and convex for x >__O. Let the set (y) of n real numbers be given except in arrangement. Then n
(5)
~'f(lY,~=l
Y,+I 1)
(where Yn+ i = Yi) is minimal if (y) is arranged in circular symmetrical order. Moreover, if the convexity o f f ( x ) is strict and no three elements of (y) have the
24
ALFRED
LEHMAN
[March
same value, then (5) attains its minimum only if (y) is arranged in circular symmetrical order. Proof. (5) is clearly invariant under all circular rearrangements. The first assertion of the theorem is thus equivalent to
~ f ( I Y , - Y , + I ] ) > ~ f(IYF
(6)
i=1
i=1
-y7+~]). (y,.+l=Yl,
Yn'-+l= Yl),
where ( y - ) = (y~,...,y~)is the symmetrically decreasing arrangement of (y) satisfying (1'). To prove the second assertion of the theorem we have to show that equality holds in (6) only if (y) is of circular symmetry. The proof proceeds by induction. For n = 2 and n = 3 every set is of circular symmetry and (5) is clearly invariant under all rearrangements, hence the theorem holds trivially for those n. We now assume the validity of the theorem for sets of n - 1 numbers and show that this implies its validity for sets of n numbers. Let (y) = (Yl,"', Y,) be such a set. Without loss of generality we may assume that (7)
0 = Yt <-- Yi,
i = 2, ...,n.
(1') and (7) imply that for ( y - ) = (y ~-, .-., y~) (8)
0 = y~- < YT,
i = 2,..., n.
We define the set (x) = (xl, "",xn-1) of n - 1 numbers by (9)
x~= y~+l,
i = 1 , . . . , n - 1,
(x~ >= 0).
Similarly, (x') = (x~, ---, x',_ 1) is defined by (10)
x'i = Y~-+1,
i = 1,-.., n - 1,
(x[ > 0).
(7) and (9) imply
~-,f(ly,-y,+xl)=
(11)
i=1 n-1
= ~ : f ( I x , - x , + l I) i=1
i=1
+f(xl) + f ( ~ - l ) -f(Ixl-xn-i [),(y.+l= y , - - 0 ; x~=xO.
Similarly, it follows from (8) and (10) that n-I
~f(ly7
i=I
- y~l])
= Z f(lxl - xi+~ l) +f(x;) +f(x~_ ~)-f(lx~
(y;+~= We define
- x'_t
i=I
y/- -- o; x,', = xD.
l),
A RESULT ON REARRANGEMENTS
1963]
s>0,
g(s,t) = f ( s ) + f(t) - J ( l s - t I);
(13)
25 t>0.
(11)-(13) give
Z f ( I y , - y,+ll)- i ~= 1 f ( l y ; - y-+~t)
i=1
(14) n-1
= Z f(lx,-
n-I
x,+~ 1) -
i=1
Ef(lx"
-
x'+~ l)
g(xl,x,,_O - g(xl,x'-O.
+
i=l
(7)-(10) imply that (x') is a rearrangement of(x). (1') and (10) give (15)
X n' - 1
<
=
'_x.
X1 --
n -2
<~--. . . <
t ~-- X [ n [ 2 ]
•
Hence, by (2), (x') is of circular symmetry (and indeed (x') = (-x)) and it follows by the assumption of the induction that n-1
(16)
]~f(lx,-x,+,[)
n-1
> ]~f(lx~-x;+,[),
i=1
(x,=xa,
x',=x'O.
i=1
f(x) is, by assumption, non-decreasing and convex for x > 0. This implies (and is equivalent to the fact) that g(s,t)= g(t,s), defined by (13), is a non-decreasing function o f s and t, s > 0, t > 0. (15) shows that x',-i and X'l are the two smallest numbers of (x). It follows that
g(xl, x._ 1) > g(x[~ x'_ 1).
(17)
(14), (16) and (17) imply (6) and we thus proved the first assertion of the theorem. To prove the second assertion we assume that ( y ) = (Yl,"',Y.) takes every value at most twice. It follows that the same holds for ( x ) = (xa,-..,x._l) and that this set takes the value 0 at most once. We also assume that the convexity of f(x) is strict (and it thus follows that the non-decreasing function f(x) is strictly increasing). This implies that g(s, t) is a strictly increasing function of s and t(s > O, t > 13); g(s, 0) = g(0, t) = 0; g(s, t) > O(s > O, t > 0). Assume now that equality holds in (6); i.e. (6')
~f([Y,-Y,+I[) i=I
= ~f(lY~--Y~x[),
(Y,+t=Yl=Y,'~+I=Y-I
=0).
i=1
(6'), (14), (16) and (17) i m p l y n-1
(16')
n-1
Z/(Ix,-x,+ll)= Z/(Ixl-x;+,l), i=l
(x.=xl, x'--x~)
i=1
and (17')
g(xl, x._ 1) = g(x~, x'_ 1).
As shown above, the rearrangement (x') of (x) is of circular symmetry. (16') ira-
26
ALFRED LEHMAN
[March
pries thus by the assumption of the induction, that (x) itself is of circular symmetry. x~ and x',_ 1 are (by (15)) the two smallest numbers of(x'). (17') and the properties of g(s, t) imply therefore that either a) xx and x , _ 1 are both positive and are the two smallest numbers of (x); or b) either xi or x,_ ~ is equal to zero, but not both. At this point it is convenient to use (9). The set of n - 1 non-negative numbers (Y2, "", Y,) is of circular symmetry and does not take any values more than twice. In case a) Yz and y, are the smallest numbers of this set and they are positive. It follows that this set satisfies (18)
(0 < )yz <= y, < Y3 < Y,,-i < "'" < Y[t,,+3)/2
or
(19)
(0 < )Yn <= Y2 <~ Yn-t -------Y3 =< "'" -<- Y[(n+ 2)/2]"
(Note that we used the fact that the smallest value is taken at most twice.) In case b) we know that either Y2 ----0 or y,, = 0 but not both, and that no other number of the set (Y2, "",Yn) of circular symmetry vanishes. Hence, one of the following four cases has to occur: (18')
(0 = )Y2 < Y, <=Y3 <-- Yn-1 <= "'" <= Y[(n+3)/2],
(20)
(0 =
(t9)'
(0 = ) y ,
(21)
(0 = )Yn < Yn-1 ----
)Y2 < Y3 =< Yn ----
If we now complete the set (Y2, "", Yn) to a set of n elements by adding the term Yt = 0 in the first place, then it is easily seen that in each case also the new set (Yl,'",Y~) is of circular symmetry. (For (18) and (18') see (3); for (19) and (19') see (1); for (20) see (4), and for (21) see (2)). This completes the proof of the theorem. We now show that all assumptions of the theorem are necessary. Consider e.g. the sets (22)
(1, 3, 4, 2) and (1, 2, 3, 4).
Here (and also in (23) and (24) below) the first set is of circular symmetry but the second is not. The sum (5) becomes, respectively, 2f(1) + 2f(2) and 3f(1) +f(3). As there are increasing functions for which 2f(2) > f ( 1 ) + f ( 3 ) we cannot drop the assumption of convexity. Similarly, for the sets (23)
(1,3,5,4,2) and (1,2,4,3,5),
The difference of the sums (5) becomes f(2) - f ( 4 ) , and we thus need the assumption t h a t f ( x ) is increasing. Using the sets (22) and the functionf(x) = x, it follows
1963]
A RESULT ON REARRANGEMENTS
27
that we have to assume strict convexity for the second part of the theorem. Finally for the sets (24)
(1,2,3,4,2,2) and (1,2,2,3,4,2),
the sums are, for a n y f ( x ) , equ~.l; hence it is necessary to assume that no three elements of (y) have the same value. An analysis of the proof shows that if in a set (y) -- (Yl,'", Y,,) one or several values are taken more than twice, then - - under the stricter assumption on f ( x ) - (5) becomes minimal for exactly those rearrangements of (y) which are constructed by the following rule: Let (z) = (z 1,'", Zm), m < n, have the same elements as (y) except that, if (y) takes a value more than twice, then (z) contains only two elements having this value. Arrange (z) in circular symmetrical order and obtain the minimizing rearangements of (y) by inserting the additional n - m elements next to elements having the same value. The theorem yields the following result concerning linear arrangements. COROLLARY. Let the function f(x) be non-decreasing and convex for x>O. Let the set (y) of n non-negative numbers be given except in arrangement. Then n--1
]~ If(Y,) + f(Yi+ 1) - f ( [ Y ,
(25)
-
Y,+ a I)]
i=1
is maximal if (y) is arranged in symmetrical decreasing order. Moreover, if the convexity o f f ( x ) is strict and if all the elements of (y) are positive and no three of them have the same value, then this maximum is attained only if (y) is symmetrically decreasing. ProoL n-1
['f(Y') +f(Y'+ ,) - f ( l Y , - Y,+ i I)] i=l
(26) n
=
2 ~ f(y,)
II
-
i=1
~] f(lY,
- Y,-I
l) - [f(Y,)
+f(Y.)
i=1
-f(lY~
- Y.
l)].
" lf(Yi) i s invariant for all rearrangements of (y), (13) and (26) give As ~i= n-1
n-i
2 g(YT,YT+l)- ~, g(Yi, Yi+l) i=l
i=1
(27) =
~ f(lY,-y,+l l) -
i=1
~f(lYY - y[+l l) + g(yl,y,) -
i=1
y~- and y~- are (by (1')) the two smallest elements of(y); hence
g(yly'~ ),
28
ALFRED LEHMAN
g(y~, y.) > g(y'~, y~).
(28) (6), (28) and (27) imply n-1
(29)
n-1
n-1
~, g(Yi, Yi+l) < ~, g(YT,Yi'+x)= ~, g(-Yi,-Yi+I),
t=1
i=1
i=1
and thus we have proved the first assertion of the corollary. If y~ > O, i = 1, ..., n, then it follows from the stricter assumptions onf(x) that
g(ya, y,) = g(y~, y; )
(28')
implies that Yl and y. are the two smallest elements of (y). By the second part of the theorem it also follows that (6')
~ , f ( l y , - y , + , l ) = ~ f(]yi-yi-+al)
i=1
i=1
implies that (y) is of circular symmetry. Under the assumptions of the second part of the corollary it thus follows (using also (6), (28) and (27)) that (25) becomes maximal only if (y) is of circular symmetry and if y, and y. are its two smallest elements. But this implies (y) = ( y - ) or (y) = ( - y ) and the proof of the corollary is complete. We now show also for the corollary that all its assumptions are necessary. Note that in (22) and (24) (and also in (30) and (31) below) the first set is symmetrically decreasing but the second is not. For the sets (22) the sum (25) becomes respectively, - f ( 2 ) + 2f(3) + 2f(4) and - 2f(1) + 2f(2) + 2f(3) + f(4). As there are increasing functions for which f ( 4 ) + 2 f ( 1 ) < 3f(2) we cannot drop the assumption of convexity. For the sets (30)
(1, 3, 2) and (3, 1, 2)
the difference of the sums (25) becomes f ( 3 ) - f ( 1 ) , and we thus need the assumption that f(x) is increasing. Using again the sets (22) and the function f(x) = x, it follows that we have to assume strict convexity for the second part of the corollary. The sets (24) prove again that it is necessary to assume that no three elements of (y) have the same value and, finally, the sets (31)
(0,2, 1) and (0, 1,2)
show that we have to assume Yi > 0 for the second part of the corollary. REFERENCES 1. Hardy, G.H., Littlewood, J.E. and P61ya, G., 1952, Inequalities (2rid edition) Cambridge. 2. Schwarz, B., 1963, Bounds for the principal frequency of nonuniformly loaded strings, Israel Jour. Math., 1, 11-21. MATHEMATICS RESEARCH CENTER, U.S. ARMY, MADISON, WISCONSIN
AND RENSSELAERPOLYTECHNICINSTITUTE,TROY, N.Y.
T A U B E R I A N THEOREMS BY
A. M E I R ABSTRACT
Tauberian constants and estimates are calculated for the difference of two linear transforms from the form (1.1) of the same function satisfying Tauberian conditions. Applications for number-sequences, and connections with previous results are shown.
1. Introduction. Denote by T(y) (y > 0) some integral-transform of a function J(x) ( - oo < x < + ~ ) of the form (1.1)
Ff(x)d(1
T(y) - Tp(y) =
- fl(y - x))
where fl(x) is a function of bounded variation on - ~ < x < + oo. In addition to Tauberian theorems, which give information about lim~f(x) if limy T(y) exists, it is possible to find estimates concerning [ Tp(y) - J ( x ) I even if neither the existence of limy T(y) nor that of lim~f(x) is assumed. Studies of such problems concerning number-sequences and the usual Abel-transform originated with the paper of Hadwiger [5], concerning integral-transforms with Agnew [2], Delunge [3], Rajagopal [10], Jakimowski [8]. In Section II of the present paper we shall obtain a simple p r o o f of estimates of i Tp(y) - T~(~l) I where Tp and T~ are transforms satisfying certain general conditions, y and 17 tend to + oo with a connection on y - I/. Our result contains as special cases many known results. 2. Tauberian Constants. The main result is the following: THEOREM 1. Let f ( x ) ( - - ~ < x < + oo) be a real or complex-valued, continuous, almost everywhere differentiable function satisfying lim f ( x ) -- 0
(2.1)
x.~
(2.2)
- oO
f ' ( x ) = 0 ( 1 ) for - oo < x < + x
(2.4)
lim If'(x) I = L < + co. X-.-~ -b o0
Received February 10, 1963.
29
30
[March
A. MEIR
Let fl(x) and y(x) be real functions of bounded variation over - oo < x < + 0% satisfying (2.5)
lim fl(x)= lim ~,(x)=0, X - ~ - - O0
lim f l ( x ) =
X - ~ - - O0
/~(x) • L~ ( - co, o) (2.6)
;
(1 -- ~(x)) e LI(0, + co)
[ ( 1 - fl(x)) e Lx(O, + co);
fo
ld#(x) ldu < + oo,
--00
X"-~ -I- O0
y(x) e L 1 ( - co, 0)
1
(2.7)
lim ~,(x)= 1
X ' ~ % O0
--QO
f f. O0
Idr(x) ldu < + co --00
and let y = y ( t ) and r/=r/(t) be positive increasing function with limt_,~oy(t) = + co, limt~ooq(t) = + oo and lim (y(t) - ~l(t)) = q ,
(2.8)
- co < q < + co
t ' - * + O0
then
li---m- [ Tp(y) - T~(n) ] < L- A~
(2.9)
t~+oO
where (2.10) and Aq is the best possible constant satisfying (2.9), since there exists a real function f ( x ) , with 0 < L < + co and such, that in (2.9) the equality sign holds. Before giving the proof we mention some particular examples. EXAMPLE (i). Let {Sk} (k > 1) (Sk = al + a2 + ... + ak) be a real or complex sequence satisfying k a k = O(1), and define (2.11)
fl(x)=e -e-~,
7(x)= {01
x - log n
(2.12) f ( x ) = k = l a k
+a,+llog(n
+ l)_logn,
x~0X<0
log n < x < log(n + 1)
If we denote e " = z, e-e-" = p, then it is easy to see that TT(q)= st, 1 + o(1) and Ttj(y ) = (1 - p) ~,kZ 1Skp k + 0(1). Thus we have by Theorem 1 lim [st, l - A(p)] < L t~
-I- o o
e-e-Xdx + -
_ e-e-x)d x -log
1963]
TAUBERIAN THEOREMS
31
where L--limk_.oo [ karl, A(p) is the usual Abel-transform o f the series {Sk}, Z ~ + ~ , p --~ 1, Z(1 -- p) ~ q as t ~ + oo. This is an equivalent form of a result o f Agnew [7]. (ii). Let {Sk}, f(x), )'(X) and L defined as in example (i). Let for 0t > 0,
EXAMPLE.
(2.13)
fl(x) =
If we denote
d = u, e ~ = z,
1 - e-~)~
and f ( l o g
x) =
x
0;
s(x) we
have
T~,(r/) = s M + o(1) (2.14) TAy)
=
-
f
x)'-'dx - c(')(.)
C(~)(u) being the C~sar6-transform o f order 0t o f the function s(x). N o w by T h e o r e m 1 li--m
I st, - c(')(u)] =< L " A(~")
t.-+ O0
where ~-* + 0% u ~
+ 0% ~u-* ~ q > 0 , as t ~
~--Iog | /
+ 0% and
/D+oO
(1 - e-")*dx + |
(1 - (1 - e-~'):)dx
[ logq + f ; ( 1 - ( 1 - e - * ) ' ) d x
if q < 1
if q => 1
This is a result analogous to a theorem of V. Garten [8] and A. Jakimowski [9]. EXAMPLE (iii). The [J,f(x)]-transformations were defined in [6] as follows: Let ~b(x) be a function of bounded variation in [0,1], ~b(0 + ) = ~b(0)=0, q~(1- ) = qS(1)= 1. The [J,J(x)]--transforms of a sequence {s,} (n > 1) (s,, = at + ... + a,,) is (2.15)
F,(x) = k~,=1Sk ---~!
log
ddp(u)
If we denote x = e", u = e _ e - v , q~(u) = fl(v), we have (2.16)
f +Go ekv
F,(x) = F,(e') = k=~tSk _
v
-Ere -e d(l - / ~ ( y - v))
Now, if ka k = O(1), and B(x)l= a'Jk ~ = 1 skkrxkl1 k !)e-* is the usual Borel-transform o f the sequence {Sk}, it is easy to show (see for example my paper [10]) that
32
[March
A. MEIR
lim IB(x) - Stxl[ = 0
(2.17)
x.-+ ÷ oO
thus
B(e ~) = O(v)
as v ~ + oo.
Therefore, if fl(v) satisfies the conditions of Theorem 1, it follows by (2.7) that the integral
f_+°°B(e~)d(1- fl(y - v)) is absolutely convergent, and thus we may write (2.16) in the equivalent form
(2.18)
F~(x) = f_~°°B(eO)d(1- /3(y- v)).
Also, by (2.12) and (2.17), lim I B(e~) - f(v) I = 0 ; t,--+ + O0
therefore, it is easily seen that lim I F,(x) - Tp(y) [ = 0,
(2.19)
y~+Oo
where T,(y)
=
- B(Y - v)).
Now, if ?(x) is as in (2.11), e~= z, and L = limk-,+oo[ kak], by Theorem 1 and (2.19) lim I st,l - Fc,(x) l <=L . A(Jp, q)
(2.20)
t~+o0
where z --} + 0% x --} + o% z x - 1 _.} q > 0 as t --} + oo, and A(q~, q) =
q~(¢e-~) I dv +
[ 1 - q~(e-e- ~)[ dv log q
This is a more general result than Jakimowski's in [8], namely we do not suppose here that ~b(x) is a monotonic function of x in [0,1], but only that the conditions of Theorem 1 are satisfied concerning fl(x) = ~b(e-e-x). The monotonicity of q~(x) clearly implies our assumptions. EXAMPLE. e -e- ~ = p, we
(iv).
Let fl(x) be as in (2.13) and 7(x) = e -"-x. Then if e y = u,
have
Tp(y) = CC~)(u) T~(tl) = A(p)
as in (2.14) as in example (i)
1963]
TAUBERIAN THEOREMS
33
Now, by Theorem 1 lim I C~)(u) - A(p) l < L . A(q,~) t.-+O0
where u ~
+ m, p ~ 1, u ( 1 - p ) ~ q > 0 i f t ~
a(q,
= |,
n0
d
e_qe-x ex +
:0o0
+ oo,
I(1 - e-X)" - e-qe- l ex.
I f q < ~ the sign of absolute value in the second integral can be removed, since e - " > 1 - u for all u > 0. In particular, if e = q = l, lim I C(1)(u) - A(p) I <=L " .,11 t ~aO
where u --* + m, p ~ 1, u(1 - p) --. 1 as t ~ + m, and an easy calculation yields
AI=I--C, C being Euler's constant. Concerning transforms Tp(y) with a monotonic increasing function fl(x) we shall be able to prove the much more general result: THEOREM 2. Let fl(x) be an increasing function satisfying the conditions of Theorem 1 and let s(x)= f ( l o g x ) be a real, or complex valued slowly oscillating function for x > 0 (i.e s(x) - s ( y ) ~ O , if x ~ + m, xy -1 -~ 1). Then for every fixed q, - ~ < q < + o% (2.21)
lim If(t/) -
t~O0
Z#(y)] _<_L * .
Aq*,
where y = y(t) -~ + o% ~/= r/(t) ~ + m, y - r / ~ q as t ~ + m, (2.22)
L* =
lim
l~n~ ~,o ~.~_x+~-.oo
[f(y) -f(x) I
y- x
and (2.23)
A* =
[3(x)dx +
(1 - ~(x))dx.
Moreover, the constant A* in (2.21) is the best possible in the same sense as in T h e o r e m 1. REMARK. It is well known, that if f ( x ) satisfies conditions of Theorem 1, then s ( x ) = f ( l o g x ) is slowly oscillating and L* in (2.22)is < than Lin(2.4). By (2.23), A* has the same value as Aq by Theorem 1 for the special case Te(~/) = f(~/). Thus, for increasing B(x) Theorem 2 is a more general result. Proofs. In the p r o o f we shall need the following modification o f the lemma of Agnew [9].
34
A. MEIR
[March
LEMMA. Let H(t,x) be a real function of the variables
t,x(t > O; - oo < x < + o o ) satisfying the following conditions :_~°°[H(t,x)[dx exists f o r t > 0, lira t"-~ o0
and suppose
L
I H(t, x) I dx = 0
for every c,
~m [+°°ln(t,x)ldx=A< +~. Let g(x) be any function of the real variable x, ( - ov < x < + oo) satisfying g(x) = O(1) for - oo < x < + 0% and suppose
li--m ]g(x)
I = L < + oo.
X"-~ + O0
Then for
T(t) = f +°°H(t,x)g(x) we have (2.24)
lim
[ T(t)[
=< L . A,
t--* -I- O0
and the constant A is the best possible in the sense that there exists a real function g(x) with 0 < L < + ov such that in (2.24) both sides are equal. The proof of the lemma is the same as the very similar lemma of Rajagopal [10].
Proof of Theorem 1. By (1.1) we have
Tp(y) - Tr(rl) = f _+:f ( x )d {'y(tl - x) - fl(y - x)}; by (2.3) +o0 ~ x
=
LJ
By (2.2), (2.6) and (2.7) we may interchange the order of integrations and by (2.5) we obtain
-
:? _
f'(u)n(t,u)du,
19631
TAUBERIAN THEOREMS
35
with y = y(t), t / = r/(t). Now it is easy to check that H(t, u) satisfies the conditions of the Lemma; thus we have lim ] Tp(y) -- T~(t/) ] < L . Aq, I-** O0
where Aq=
t--~oli--mf _ + ° ° [ [ 3 ( y - u ) - 7 ( ~ l - u ) l d u ,
and by (2.8) and Lebesgue's well known theorem =
]fl(x) - ~,(x - q) ldx. --00
Q.E.D.
Theorem 2. First, by a slight modification of a theorem of R. Schmidt [11], for all x , y satisfying Ix - y] __>c~> 0 Proof of
(2.25)
If(x) - f ( y ) ] < K~.[ x - y I,
K being dependent only on 6. Thus the lim defining L* in (2.22) certainly exists. In order to prove our Theorem it is enough to show that (2.21) holds; the fact that A* is the best possible constant satisfying (2.21) follows from the remark after Theorem 2. Let now be e > 0 given; define 6 > 0, x o > 0 such, that for x > Xo, Y _>-Xo, Ix - Y] => 6 (by (2.22)) (2.26)
If(x) - f ( y ) [ < (L* + e) [ x - y l,
and for x > Xo, y > Xo and Ix - y[ < 6 (since f(log x) is slowly oscillating) (2.27)
If(x) - - f ( y ) [ < s.
Now f(tl) - Tp(y) =
(f(tl) - f ( x ) ) d ( 1 - fl(y - x)) oo
(2.28) =
Let y , q > x o + 6. By (2.25)
=0
E r; + axo
(~/-y)(1-fl(y
Jq-,~
Xo))+
f)
+~= It + I 2 + I 3 + 1 4
udfl(u) --XO
and by (2.5) and (2.6) we obtain easily (2.29)
I t = o(1)
as y -~ oo.
36
A. MEIR
By (2.26) 112 [ < (L* + e
( t / - xld(1 - fl(y - x l ) < (L* + e o
dfl(u vy--TI
dt ,J y - T i
(2.30) = (L* + e)"
(1 - fl(u))du
I n the same way (2.31)
]14] _-<
(f
)
• (L* + e).
By (2.27) /* y--t/+d
(2.32)
1131 z
dB(u)-<
Since e > 0 was arbitrary we have by (2.28) - (2.32), as t -~ o% y ~ oo, y - t/-~q, li--m If(t/) - Ta(y) [ < L * . A t t-~O0
Q.E.D. REFERENCES
1. Agnew, R.P., 1949, Abel transforms and partial sums of Tauberian series, Ann. of Math., 50, 110-117. 2. Agnew, R.P., 1952, Integral transformations and Tauberian Constants, Trans. Amer. Math. Soc., 72, 501-518. 3. Delange, H., 1950, Sur les th6or6mes inverses des proc6d6es de sommation des s6ries divergentes, Ann. Sci. Ecole Norm. Sup (3) 67, 99-160. 4. Garten, V., 1951, Uber Taubersche Konstanten bei Cesar6schen Mittelbildung, Comm. Math. Helv., 25, 311-335. 5. Hadwiger, H., 1944, Uber ein Distanztheorem bei der A-Limitierung, Comm. Math. Helv., 16. 209-214. 6. Jakimovski, A , 1960, The sequence-to-function analogues to Haussdorfftransformation, Bull. Res. Count. of lsrael, 8F, 135-154. 7. Jakimovski, A., 1961, Tauberian Constants for Haussdorff transformations, Bull. Res. Count. of Israel, 9F, 175-184. 8. Jakimovski, A., 1962, Tauberian Constants for the [J,f(x)]-transformation, Pacific Journ. of Math., 12, 567-576. 9. Meir, A., 1963, Tauberian Constants for a family of transformations, Annals of Math., (to appear). 10. Rajagopal, C.T., 1956, A generalization of Tauber's theorem and some Tauberian constants III, Comm. Math. Helv., 30 63-72. 11. Schmidt, R., 1925, ldber divergente Folgen und lineare Mittelbildungen, Math. Zeitsch., 22, 89-152. THE HEBREW UNIVERSITY OF JERUSALEM
ON SERIES IN LINEAR TOPOLOGICAL SPACES* BY
ARYEH DVORETZKY ABSTRACT
The main result is that in every complete locally-bounded linear topological space there exist series which are unconditionally yet not absolutely convergent. Relations between absolute, unconditional and metric convergence of series are studied. 1. Introduction. The primary purpose of this paper is to study the connection between absolute and unconditional convergence of series in general linear topological spaces (over the real or complex field). Since our auxiliary considerations involve metric spaces we study also the connections with metric convergence. There being no universal agreement about the terminology we explain the above terms at the outset. A series cO
(1.1)
x, n=l
in a linear topological space is said to be absolutely convergent (A) if oo
(1.2)
~, Mv(x,) < eo n=l
for every neighborhood V of the origin, where Mv(x ) is the Miukowski functional of x relative to V, i.e. (1.3)
My(x) = inf{2 : 2 > 0, x s 2V}.
A series (1.1) in a linear topological space is said to be unconditionallly, or commutatively, convergent (U)if every series obtained from (1.1) by rearrangement is convergent. In complete spaces this is equivalent to the requirement that (1.1) is sub-series convergent, i.e. that ~v% 1 x,~ be convergent for every strictly increasing sequence of positive integers nv. Received Feburary 15, 1963. * The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force. 37
38
ARYEH DVORETZKY
[March
In complete locally convex spaces A =~ U. Both these notions are topological and the properties of absolute and unconditional convergence are preserved under isomorphisms. The third kind of convergence we consider is of a different nature. A series (1.1) in a linear metric space is said to be metrically convergent (M) if oo
(1.4)
~ d(x,,O) < oo, n=l
where d(x, y) denotes the distance between x and y. This is very far from being a topological notion. Even one dimensional space can be metrized in such a way that M does not imply even ordinary convergence; or also so that A does not imply M. Nevertheless it will prove very useful for studying the relations between the topological notions A and U. If the metric in the space is translation invariant we write IIx II = d(x,O) and (1.4) becomes oo
(1.5)
z Ilxoll < oo n=l
(In general this 'norm' is not linear homogeneous. II~x 11is continuous in both variables but need not even be monotone in 2 for 2 > 0.) In Banach spaces (1.5)is equivalent to the absolute convergence of (1.1). In arbitrary complete metric spaces with a translation invariant metric (F* spaces) we still have M =~ A, but the converse need not be true. It is well known that A ¢:, U in finite dimensional spaces. C.A. Rogers and the author proved [4] that this is not the case in any infinite dimensional Banach space, i.e. in every such space there exist series which are U but not A. This is equivalent to saying that in every infinite dimensional Banach space there exist series which are U but not M. This last statement was generalized from Banach spaces to arbitrary F* spaces by S. Rolewicz(~) [8]. Since A need not imply M in these spaces it does not entail the existence of series which are U but not A, and indeed in the space of all sequences x = (~1 .... , ~,,...) with the topology of coordinate convergence(2) A~* U (see e.g. ['1] p. 63). Our main result is that in every infinite-dimensional locally-bounded(a) complete (1) Employing an older terminology S. Rolewicz calls M absolute convergence. The author in [3] also uses the older terminology. The present paper was announced in [3] as forthcoming under a different title (On series in Fr~ahet spaces.) (2) This space can be metrized by cO
Ilxll = Z 2-"1 .1(1 n=l
(3) This means that there exists a neighborhood of the origin which is contained in a sufficiently large homothetic image of any other given neighborhood of the origin.
1963]
ON SERIES IN LINEAR TOPOLOGICAL SPACES
39
linear topological space U does not imply A. This seems a rather surprising generalization of the result about Banach spaces. In the opposite direction we show that every linear metrizable topological space containing arbitrarily shortrays(4) has an infinite dimensional subspace in which U and A are equivalent (we construct a subspace isomorphic to the one given in footnote(2)). It is remarkable that local convexity does not enter into these statements. 2. Statement of results. We begin with results about metric convergence. For a linear metric space we put A(X) = sup d(x, 0).
(2.1)
x,X
THEOREM 1. I f X is a complete linear metric space containing arbitrarily short rays and (7,,) is a sequence of positive numbers satisfying 0 < ~, < A(X)
(2.2)
(n - 1, 2,...)
and (2.3)
lim ~,
= 0
n----O0
then there exists in X an unconditionally convergent series (1.1) with (2.4)
d,,(x,0) = ?,
(n = 1,2, ...).
In order to state concisely the next results we introduce the following definitions. Let X be a linear metric space and p > 0. We put (2.5)
B*(p)=(x:d(l~x,O)<=p
for
0 _ < _ / ~ 1}.
B*(p) is a closed star-shaped neighborhood of the origin. The sets B*(p) obviously constitute a fundamental system of neighborhoods of the origin. Let if(2) be any positive function defined for 0 < 2 < 1. We say that X has property C*(~b) if (2.6)
B*(2p) ~ qS(2)B*(p)
for all p > 0 and 0 < 2 < 1.
The following may be considered the key result of the paper. THEOREM 2. Let X be an infinite-dimensional complete linear metric space having the property C*(dp) and let ~,(n = 1,2, ...) be a sequence satisfying (2.2)
and oO
(2.7)
]~ ~2(a~,) < oo
for every 0 < o- < m .
n=l
(4) This means that to every neighborhood of the origin there corresponds some x ~ 0 for which the whole ray 2x, (2 > 0), is contained in it.
40
ARYEH DVORETZKY
[March
Then there exists in X an unconditionally convergent series (1.1) for which (2.4) holds. An immediate consequence is the following COROLLARY 1. Let X be an infinite dimensional complete linear metric space with a translation invariant metric and let yn(n = 1, 2.... ) be a sequence satisfying (2.2) and (2.8)
~ ~ < m.
Then there exists in X an unconditionally convergent series (1.1) with
IIx. I] = Y.,
(2.9)
(n = 1,2 .... ).
Indeed, this result follows at once from the observation that if the metric is translation invariant then X has the property C*(q~) with ~b(2) = 22. This is seen as follows: If x~B*(2p) with 0 < 2 < 1, then for every 0 < # < 1 we have(5)
=
[l]j 2- .[1N --<[11ap = p,
!.e. (x/2,l) e B*(p) as asserted. For Banach spaces this corollary is precisely the main result of [4]. It is again noteworthy that local convexity is not needed. Taking e.g. y, = 1 / n (for n > 1/A(X)) we obtain the result of S. Rolewicz quoted in the introduction. As another consequence of Theorem 2 we deduce our principal result of a topological nature. Tm~OREM 3. In every infinite-dimensional locally-bounded complete linear space there exist unconditionally convergent series which are not absolutely convergent. In the converse direction we prove
Trmo~M 4. Every metrizable complete linear space containing arbitrarily short rays has infinitely dimensional subspace in which every unconditionally convergent series is absolutely convergent. We conclude with a result showing how little connection there is between M and A even in F* spaces. THEOREM 5. Let X be a metrizable linear topological space and y,(n = 1,2,...) any sequence of posit&e numbers satisfying (2.3). Then there exists a translation invariant metrization of X (reproducing, of course, the given topology) such that (1.1) is absolutely convergent whenever (5) Square brackets denote the integral part.
1963]
ON SERIES IN LINEAR TOPOLOGICAL SPACES IIx, II <=3',,
(2.10)
41
(n = 1, 2 .... ).
3. Proof of Theorem 1. Given e > 0 there exists fl > 0 such that d(x,O) < fl and d(y, 0) < fl together imply d(x + y, 0) < ct. Thus starting with any % > 0 we can define consecutively a sequence of positive numbers el, e2 .... , c~v.... satisfying (3.1)
d(x + y,O) < ~ - 1 whenever d(x,O) <=ct, and d(y,O) __
(v = 1,2,...).
We may moreover assume (3.2)
lim 7~ = 0. v~O0
(This is automatic if % < A(X), then ~v < 2-V~o). Put (3.3)
d(2x, O).
A(x) = sup .I>0
Since X contains arbitrarily short rays there exist non zero yv in X satisfying (3.4)
A(y~)<e~
( v = 1,2 .... ).
Finally we determine a strictly increasing sequence of positive integers k, such that (3.5)
7,
for
n>k~
( v = l , 2 .... ).
We now define x n satisfying (2.4) as follows: Arbitrarily for 1 __ki m
(3.6)
j-1
]E' x. = n =k i
kv+l-1
]E v =~
~mt
]E' x. + n =k v
x. n =kj
where j is determined by kj < m < kj + 1. By our construction, (3.3)and (3.4), we have (3.7)
d
x,,0
_-< A(y~)
d
x,,0
_<-A(y/)<~j.
ll=kj
n=kv
Applying (3.1) from the last summand of (3.6) backwards we obtain d
~ ~-l. /I--
As this holds for any m => k s it follows, by (3.2), that ]E'x, is Cauchy convergent; hence, by completeness, convergent. This being true for any subseries, the theorem is established.
42
ARYEH DVORETZKY
[March
4. A geometricallemma. LEMMA 1. Let B be a compact set in Euclidean m-space containing the origin as an interior point. Let 1[ [I denote the Euclidean norm, relative to a given orthonormal set et .... ,e,,. There exists a centro-affine transformation T having the properties (4.1)
{x : Ilx Ii < 1} = T B
and there are points Pl ..... p., on the boundary of T B such that J
(4.2)
pi = ]g rtj,~ e~
(j = 1 , . . . , m )
t=1
with j--1
(4.3)
2
2
rtj,~ = 1 - nj,j =< j - 1
i=l
m
Specialized to convex symmetrical B this is precisely the fundamental lemma of [4]. The proof, however, does not appeal at all to the symmetry or convexity of B and holds verbatim for any compact B containing the origin as an interior point (it is even possible to relax somewhat the requirement of compactness). Since the proof is reproduced in Day's book [1] (p. 61) we shall not repeat it here. For any real 21 .... ,2 k (1 < k < m) we have k
k
k
E 2~pj = Y. 2Fcj,~e ~ + Y_. ;q(p~ - zj,jei).
(4.4)
j=l
j=l
j=l
By (4.2) and (4.3)
~lk
]IjX=I
(4.5)
(
)1/2
!i --< i=~ ~ 2~
and (4.6)
iIjZ__xAj(p~-n~,~ej)
< Z j=l
12 1
_-< • 22
j - 1 ]
\j=l
\j=l
m
/
"
Combining (4.4), (4.5) and (4.6) we obtain(6) (4.7)
~= 2jp i j
I
< =
1+
/ ~ 22} 2m
\j=l
"
/
Taking this into account we may reformulate the above lemma as follows: (6) A similar computation leading to a somewhat weaker estimate occurs in [4] (and [1]). The improvement is of no consequencefor the present paper and is brought for future reference. An occasionally better estimate is given by A. Grothendieck [5] but there is a mistake in his derivation (since the exact nature of the estimate is of no importance in [5] none of its results are affected).
1963]
ON SERIES IN LINEAR TOPOLOGICAL SPACES
43
LEMMA 2. Let B be a compact set containing the origin as an interior point in real m-dimensional space. Then a Euclidean norm II II m a y be introduced into the space so that (4.8)
{x:
II x 11-<- 1} c
B
and there exist points Pl ..... p,, of unit norm on the boundary of B f o r which (4.7) holds f o r all real 21 ..... 2k (1 < k < m).
5. Proof of Theorem 2. In view of Theorem 1 we may assume that X does not contain arbitrarily short rays. Then, cf. (3.3), (5.1)
6 = inf A(x) > 0 ~X
and we may start with Xo < b and construct a sequence of positive numbers ~i, e2 ..... e ..... satisfying (3.1). Let 0 < p~ < el and let kl be such that oo 1 Z ~2(~./pI) < -.
. =k,
4
Having defined p, and kv (v = 1,2, ..,) we choose Pv+l and then k,+~ > k~ so that (5.2)
O
and
N
~z
?~
<_~.
n=kv÷l
The existence of such k~ is assured by (2.7). We now proceed to define the x. in (1.1). For 1 < n < k 1 we choose them arbitrarily subject to (2.4). For kv < n < k~ + 1 we proceed as follows: Put k = k, + 1 - k~ and let Y be an m = k 2 dimensional subspace of X . B = Y N B*(p,) is compact (since p~ < 6) and contains the origin as an interior point. Let H be the Euclidean norm in Y and pj the points whose existence is asserted in Lemma 2. Then we have
II
(5.3)
j~12jpj
<
1 +
for all real 2, ..... 2k. The points pj may not quite do for our purpose. It may happen that though they are boundary points of B still/~pj ¢ B for some # > 1 (there may be protruding segments). However, as B is compact and star-shaped there exist qj on the boundary of B, arbitrarily close to the respective pj, for which i.tqjeB for all p > 1. In view of (5.3) the qj may be chosen close enough to be pj so that we have k
(5.4)
for allreal:q,...,2k.
/
k
\1/2
j=~12,q, < 2 tj~=12:)
44
ARYEH DVORETZKY
[March
We now choose x~ for k v ____n < kv+ 1 so that it is a positive multiple, x n = psqj say, of qn-kv + 1 on the boundary of B*(7,,) and that (2.4) holds. Then we have by (2.6) (5.5)
,tj<~b ( 7 - - p ~ v ) ( j = l .... , k , , + , - k ;
n=k, +j-l).
From (5.2), (5.4) and (5.5) we obtain kv+t-I
~
x,, < 1
n
( v = l , 2 .... )
..
where' denotes summation on any subset. By (4.8) and (5.2) we have then
d
~;
x~,O
( v = l , 2 .... ).
n=kv
This is exactly (3.7) and the proof is achieved in the same way as that of Theorem 1. 6. Proof of Theorem 3. We start with a second corollary of Theorem 2 which is of independent interest. COROLLARY 2. Let X be a complete infinite-dimensional metric space with a p-homogeneous norm (0 < p < 1), i.e. satisfying (6.1)
~2xll = l~lPilxll
for all scalars ~.
Then, given any sequence of positive numbers 7n (n = 1, 2,...) satisfying (6.2)
~ 72/" < oo, n=l
there exists in X an unconditionally convergent series (1.1)for which (2.9)holds. Indeed, (6.1) implies B*(2p) = 21/PB*(p) for 2 > 0. Thus X has property C*(qS) with ~b(2) = 21/p and (6.2) is equivalent to (2.7). On the other hand, taking V = B*(p) we have, by (1.3) and (6.1),
M,,(x)
=
(ll x II/p)''.
Therefore oo
(6.3)
II x,,lt 1,p < n=l
is a necessary and sufficient condition for the absolute convergence of (1.1). It follows that in every complete infinite-dimensional space with a p-homogeneous norm there exist series which are unconditionally yet not absolutely convergent (take e.g. 7, = n -p (n = 1,2 .... )).
1963]
ON SERIES IN LINEAR TOPOLOGICAL SPACES
45
The theorem now follows from the fact that the topology of a locally bounded space can always be given by a p-homogeneous norm with a suitable 0 < p < 1. (S. Rolewicz [7], see e.g. I-6] p. 165). 7. Proof of Theorem 4. It may be assumed without loss of generality that the topology is given by a translation invariant metric. Let Yl be an arbitrary non-zero point of the space, and having determined y~ chose Y~+I # 0 so that (7.1)
1
A(yi+a) < TIIY'II,
(i =
1,2 .... ).
Given any sequence of numbers ~ ( i = 1, 2 .... ), it follows from (7.1) that the series
~, (~Yi
(7.2)
i=1
is metrically, hence absolutely, convergent. Therefore, by completeness, it represents a point in the space. Let Y be the totality of points representable by series (7.2). It is obviously a linear set and the representation (7.2) isunique. Indeed, we claim that if (7.2) has the value zero then all (~ vanish. Assume (~ = 0 for i < j, (j > 1); if (g # 0 then
1 ~,(,yi=yj+
~,
(,
contradicting (7.1). Thus ~ = 0 (i = 1,2 .... ). Moreover, Y is a closed subspace. Indeed, let oo
(7.3)
z, = •
~,,~Yi (n = 1,2,...)
i=1
be a sequence of points of Y converging (in the original space) to z. The z, form then a Cauchy sequence and, as in the proof uniqueness above, it follows that each sequence ~1,i, (2 ~..... ~,,,~... is a Cauchy sequence. Hence limn = co~,,~ exists (i = 1,2 .... ). If we denote this limit by ~ then it follows immediately from (7.1) that the sequence (7.3) tends to the point represented by (7.2). Hence z z Y and Y is closed, therefore also complete. We have in fact shown that a sequence (7.3) is convergent if and only if all the sequences (1,i, (2,i .... ,(.,i .... (i = 1,2 .... ) are convergent. Therefore, a series (1.1) with cO
(7.4)
x, = Z
~,,~Yi
i---1
is unconditionally convergent if and only if all the series
46
ARYEH DVORETZKY
[Ma rc h
oo
(7.5)
~: ~. i,
(i = 1,2 .... )
tl=l
are unconditionally, hence absolutely, convergent. But if the series (7.5) are absolutely convergent it follows that the Minkowski functional (1.3) with
v -- {x: IIx 11--- IIy J Ill satisfies co n=l
oo
j
~ I ~.,, I!1 Y, II < ~"
Mv(x.) S ~ n=l
i=l
II 11-~
Since Yi 0 by (7.1) it follows that the unconditional convergence of (1.1) implies its absolute convergence and the theorem is established. 8. Proof of Theorem 5. Since X is metrizable we can define its topology by a translation invariant metric d'. We now introduce another metric d, equivalent to d', having the required properties. Let g(t) be strictly increasing and continuous for 0 < t < ~ with g(0)= 0 and such that oo
(8.1)
~ ?.g(~,) < ~ . n=l
Put f(t)=tg(t). Then f(t) is continuous and strictly increasing from 0 to ~ and we have for 0 < s, t < s t
g(s + t)> max(g(s),g(t)) > - - - g ( s ) =
=
+ ----g(t),
s+t
s+t
or
(8.2)
f(s + t) >_f(s) + f(t) ;
and (8.2) obviously remains valid when either s or t, or both, vanish. L e t f -1 be the inverse function o f f and define
a(x, y) = f - l ( a ' ( x ,
y))
for all x, y ~ X. Then d(x, y) is again translation invariant and induces the same topology as d'(x,y). But IIx. II --<7. implies IIx. I1' _-J(r.)- ~',g(?,,). Thus, by (8.1), (2.10) implies the metric convergence of (1.1)--relative to hence its absolute convergence.
II II'--and
9. Remarks. 9.1. The requirement of completenes can be dropped throughout if unconditional convergence is replaced by Cauchy unconditional convergence (i.e. the partial sums of every rearranged series from a Cauchy sequence). 9.2. The contraction condition C*(q~) in Theorem 2 can be weakened to the
1963]
ON SERIES IN LINEAR TOPOLOGICAL SPACES
47
following one: There exists a null-sequence of positive numbers PN (N = 1,2 .... ) and corresponding to each PN subspaces YN of arbitrary high finite dimension such that YN (~ B*(2PN) c YN n ~b(2)B*(pN) for N = 1,2 .... and 0 < 2 < 1. Indeed the only modification in the proof is that the p~ in (5.2) have to be chosen from the given sequence and the m-dimensional Y from the corresponding subspaces. 9.3. Further results can be deduced if one couples C*($), with conditions of the type B*(~p) ~ ~b(;OB*(p). (A similar remark applies to 9.2 as well as to further weakenings of C*(~b)). This is particularly striking when B*(Ap) = dp(2)B*(p) as in the case of Corollary 2. Then the gain due to the second-power of 2j in (4.7) is most evident. 9.4. If I] I[ is the ordinary norm in nilbert space and we introduce a phomogeneous norm by llxH' = Ilxll p ( O < p < l ) we see from the fact that xll xo < ~ is necessary for unconditional convergence of (1.1) in Hilbert space that the result of Corollary 2 is best possible.
REFERENCES 1. Day, M.M., 1958,NormedLinear Spaces, Ergeb. d. Math., N.F., H. 21, Springer, BerlinG~ttingen-Heidelberg. 2. Dvoretzky, A., Some near-sphericity results, Proc. Symposia in Pure Math., vol. 7. 3. Dvoretzky, A., 1961, Some results on convex bodies and Banach spaces, Proc. International Syrup. on Linear Spaces, Jerusalem 1960, pp. 123-160. Israel Academy of Sciences and Humanities, Jerusalem. 4. Dvoretzky, A. and Rogers, C.A., 1950, Absolute and unconditional convergence in normal linear spaces, Proc. Nat. Acad. Sci. U.S.A., 36, 192-197. 5. Grothendieck, A., 1953, Sur certaines classes de suites dans ]es espaces de Banachet ]e th6or~me de Dvoretzky-Rogers, Bol. Soc. Mat. Sad. Paulo, 8, 83-110. 6. K6the, G., 1960, TopologischeLineare Riiume I. Springer, Bcrlia-Gottingen-H¢idelberg. 7. Rolewicz, S. 1957, On a certain class of linear metric spaces, Bull. Acad. PoL Sci., C1. III, 5, 471-473. 8. Rolewicz, S., 1961, On a generalization of the Dvoretzky-Rogers theorem, Colloq. Math. 103-106. THE HEBREW UNIVERSITYOF JERUSALEM
QUOTA GAMES WITH A CONTINUUM OF PLAYERS BY
BEZALEL PELEG ABSTRACT
In this paper the notion of m-quota game with a continuum of players is defined and the theory of bargaining sets is generalized to this new class of games. We discuss only the bargaining set M0 and our results are similar to those obtained in the finite case. Our main result is that for maximal coalition structures the stable payofffunetions are exactly those in which almost every non-weak player receives no more than his quota and the weak players receive zero. In this paper the results that were obtained in [6] for finite m-quota games are generalized to m-quota games with a continuum of players. So this work continues both the study of bargaining sets and of games with a continuum of players. The results that we obtain are somewhat sharper than those in [6], and the proofs are shorter. For an introduction to the subject o f games with a continuum of players we refer the reader to [1], which contains also a complete bibliography. 1. Definitions. In this section we define the bargaining set Mo of m-quota games. Our definitions were inspired both by those in [5] o f the bargaining set Mo of finite m-quota games(I), and by the definitions of characteristic function games with a continuum of players in [3] and [7]. We denote by I the unit interval [01]. I is our set of players. A coalition is a Lebesgue measurable subset of I. A characteristic function is a non-negative real function v defined on the set of all coalitions of I. A game is fully described by its characteristic function. Lebesgue measure will be denoted by p. DEFINITION 1.1. V is a characteristic function o f an m-quota game, 0 < m < 1, if there exists a measurable function w such that
v(s)_ffsWd, l0
,
/~(S) = m #(S)
~ m
w is the quota of the game; if it exists then it is unique (up to equivalence). The m-quota game with the quota function w will be denoted by (re,w). A weak player t is one f o r w h o m w(t) < O. Received February 15, 1963 (1) Which are a modification of the m-quota games of Kalish [4].
48
QUOTA GAMES WITH A CONTINUUM OF PLAYERS
49
Let (m,w) be an m-quota game .A coalition structure (c.s.) is a finite set of disjoint coalitions of I, whose union is I. A c.s. is maximal (m.c.s.) if it contains a maximal number of m-coalitions (i.e. coalitions whose measure is m). If b is a m.c.s, we denote by s(b) the number of the m-coalitions of b whose intersection with the set o f weak players has measure 0. DEFINITION 1.2. A coalitionally rational payoff configuration (c.r.p.c.) is a pair (x, b), where b is a c.s. and x (to be thought of as a payoff vector) is a measurable function that satisfies:
f xd# = v(B), B ~ b , and Ssxd# >=v(S), for all coalitions S, S c B ~ b. If b is a c.s. and S is a coalition we denote by P(S, b) the set of partners of S in the c.s. b, i.e. the set U{B : B eb, #(S ca B) > 0}. DEFINITION 1.3. Let (x, b) be c.r.p.c, and K and Ldisjoint, non-null coalitions with the same partners. An objection of K against L in (x, b) is a c.r.p.c. (y,c) that satisfies: #(P(K,c) ca L) = O, y(t) > x(t) for almost every t e K and y(t) > x(t) for almost every t e P(K,c). DEFINITION 1.4. Let (x, b) be a c.r.p.c, and (y, c) an objection of a K against an L in (x, b). A counter objection of L against K is a c.r.p.c. (z,d) that satisfies: I~(K - P(L, d)) > O, z(t) > x(t) for almost every t ~ P(L, d), and z(t) > y(t) for almost every t E P(L, d) ca P(K, c). A c.r.p.c. (x,b) is stable if every objection in (x, b) can be countered. The bargaining set Mo is the set of all stable c.r.p.c.'s. The following two lemmas are not difficult to prove. LE~tA 1.5. Let u(t) be a real measurable function on a coalition S. If 0 < 0 < #(S) then there is a coalition T c S such that /~(T) = 0 and inf{u(t): t E T} __>sup {u(t): t ~ S-T}. LEMMA 1.6. Let (x,b) be a c.r.p.c., K = {t:w(t)> x(t)} and L = {t:x(t)> max(0, w(t))}. If a coalition K1 c K has an objection (y,c) against a coalition S such that f l~(w - y)d# + f ~ ( w - x)dl~ < f S~L(X -- w)d#, where K2 = {t :t ~ P(K 1, c), w(t) > y(t)} and K 3 = K - P(K1, c),then S has no counter objection.
2. Stable payoff configurations. THEOREM 2.1. Let (m,w) be an m-quota game, ba m.e.s, and (x,b) a e.r.p.c. If x(t) < max(O, w(t)) a.e. then (x, b) e M o.
50
BEZALEL PELEG
[March
Proof. We have to show that if (y, c) is an objection of a coalition K against a coalition L, then L has a counter objection. We denote A = {t:w(t)<0}, Lt=LnA and L 2 = L - L ~ . # ( L 2 ) = q m + r , O < r < m . Let L3 be a subcoalition of L2 whose measure is r. Let L 2 - L 3 form q m-coalitions with a quota split; so if r = 0 L has a counter objection. I f r > 0 let U e c be an m-coalition; we assert that there is a sub-coalition U1 c U such that p ( U 1 ) = m - r , # ( K - Ux) > 0 and .fv~(W - y)d# > O. I f y(t) = w(t) for almost every t e U, then we have to delete f r o m U a sub-coalition U: c U whose measure is r and that satisfies g(U 2 ~ K) > (r/m)l~(U n K) to obtain U 1. I f S = {t:t e U, y(t) > w(t)} has positive measure, let U2 be a sub-coalition o f S that satisfies 0 < # ( U 2 ) < r/2. Denote V 1 = U - U2. f v , ( W - y ) d # > O . I f # ( K - 111) = 0 let further U 3 be a subcoalition o f 1/1 n K that satisfies 0<~t(U3) < r/2, and such that Sv2(w-y)dl~>O, where 1"2 = Vt - U3./t(K - 112) > 0. So we can always obtain a coalition V c U such that fv(w - y)dl~ > O, I~(V) > m - r and #(K - V) > 0. Let U 1 be a subcoalition of V such that # ( U t ) = m - r and inf {w(t) - y(t) : t e U1} > sup {w(t) - y(t) :t e V - Ut}. U1 has the desired properties. To complete a counter objection of L when r > 0 , La can f o r m an m-coalition F together with U1. The payments to the members of F will be w(t) for t e L 3 and y(t) + (1/m - r) ~vl(w - y)dp, for t e UI. A consequence o f Theorem 2.1 is COROLLARY 2.2. Let (re, w) be an m-quota game and b a m.c.s.; there is always a measurable function x such that the c.r.p.c. (x, b ) e M o . THEOREM 2.3. Let (m,w) be an m-quota game and b a m.c.s, that satisfies s(b) ~ 2. If a c.r.p.c. ( x , b ) e M o then x(t) < max(O,w(t)) a.e.,
Proof. 1 = mq + r, 0 < r < m. W.l.o.g. b = { B 1, ...,Bq, Bq+i}, #(B~+I) = r and # ( A n (Bq_ 1 U Ba)) = 0, where A = {t :w(t) < 0}. We denote also A1 = A - Bq+ i and R = B q + I - A . Let (x,b) be a c.r.p.c. We denote K = { t : t e B j , j < q , x(t) < w(t)}, J = {t : t e B i, j < q, w(t) = x(t)} and L = {t :x(t)>max(O,w(t))}. We shall prove that the inequality #(L) > 0 implies that (x, b) ~ M0. This will be done by proving the existence o f sets U and V such that U has an objection against 11, and V has no counter objection. We distinguish the following possibilities:
(a)
#(K) + #(J) + #(R) < m
Let T c L UA1 be a coalition whose measure is m - / x ( K ) - #(R) - #(J) and t h a t satisfies inf{w(t) - x ( t ) : r e T} ~ sup{w(t) - x ( t ) : t e ( L U A I ) - T}. P((L u A i ) - T,b) = P(K,b). We have J'xur(W - x)d# > 0. Since s(b) > 2 we
1963]
QUOTA GAMES WITH A CONTINUUM OF PLAYERS
51
have also S L - T ( x - - w ) d # > J ' T ( X - w)d#. K can object against (L u A 1 ) - T by forming, together with R u J u T, an m-coalition F; the payments to the members of F will be w(t) for t ~ R U J, x(t) for t e T and
z(t) = x(t) +
(w(t) - x(t)) fK~T(W -- x)d# J"K(W -- X) d.u
for t e K . Since f r ( W - z)d# = S K ( w - x ) d # by l e m m a 1.6 (L u A 1 ) -
(b)
fK ~ r( w -- x)d# = ~ r ( X - w)d# < ft.- r ( X - w)d#
T has no counter objection.
#(K) + p(R) < m and p(K) + #(R) + p(d) > m
K can object against L U A 1 by forming, together with R and a sub-coalition of J whose measure is m - # ( R ) - #(K), an m-coalition with a quota split. L UA1 has no counter objection.
(0
a(K) + t,(R) >_-m
p(K)+p(R)=pm+s, O<s<m. I f s = 0 K objects against L U A 1 by forming, together with R, p m-coalitions with a quota split. I f s + p(J) ~ m let Q = K be a coalition whose measure is s. K can object against L u A1 by letting K - Q f o r m together with R p m-coalitions with a quota split, and Q form an additional m-coalition with a quota split together with a sub-coalition of d whose measure is m - s. In b o t h cases L U A1 cannot counter object. So we may assume in the following that s > 0 and s + p(J) < m. We now show that we m a y also assume that #(L) > m -- s - #(J). I f #(L) < m - s - #(J) let Q c K be a coalition whose measure is s that satisfies sup {w(t) - x(t) : t ~ Q} < i n f {w(O - x(t) : t e K - Q}. B~ c P ( K - Q, b). K - Q can object against G = P ( K - Q, b) - K - J by forming, together with R, p m-coalitions with a quota split. . [ a , ~ L ( x - - w ) d # > ~ [ B ~ n z ( X - - w ) d # = f n q , ~ r ( w - x)d# > S Q ( w - x)d#, so by l e m m a 1.6 G cannot counter object. Let now S be a sub-coalition of K whose measure is s that satisfies sup {w(t) - x(t) : t e S} < inf{w(t) - x(t) : t ~ K - S}, and T a sub-coalition of L U A l whose measure is m - s - #(J) that satisfies i n f { w ( t ) - x(t) : t e T} ~_ sup {w(t) - x(t) :t ~ (L U AI) - T}. We distinguish the following sub-cases:
(c.1)
fs (w-x)d~O vT
52
BEZALEL PELEG
[March
In this case there is a coalition S 1 c K such that/ffS1) = s , P ( K - St, b)D B~uBq_ 1 and f s , ( W - x ) d l ~ < I L ~ B q u S q - , ) ( X -- w)dlt. I f P(K - S,b) ~ Bq WBq_I we may choose S 1 = S. If P ( K - S, b) gp Bq _ 1 we may assume that Bq_ 1 n K = S. Let 0 < 6 < min (s,l~(Bq n K ) ) and let U 1 c S and U 2 c K n B q be coalitions whose measure is 6. Sa = (S - Ua) W U2 has the desired properties. Now we can construct an objection of K - Sx against G = P(K - S~, b) - K - J by letting K - S x form together with R p m-coalitions with a quota split. By lemma 1.6 G has no counter objection.
f s u r (w - x)dlt > 0
(c.2)
We have that P(K, b) = P((L u A 1) - T, b) ; also f r.(x - w ) d g > 2 f T(X -- w)dlt since f T ( X - - w ) d # < IBjoL( x - w ) d # for j = q - - l , q . K has the following objection against (L w At) - T : K - S forms, together with R, p m-coalitions with a quota split, and S joins J u T to form an additional m-coalition F ;the payments to the members of F will be x(t) for t e J u T and
z(t) = x(t) +
(w(t)- x(O) Is u r ( w - x)d~ I s (w - x)d#
for t ~ S . Since IL_T(X--W) d # > J ' r ( x cannot counter object.
w)dkt = I s ( W -
z)d#(L uAt) - T
COROLLARY 2.4. Let (re, w) be an m-quota game and b a m.c.s. I f w(t)>O a.e. then a c.r.p.c. (x, b) ~ M o if and only if x(t) < w(t) a.e. To complete the p r o o f of corollary 2.4 we have to show that if s(b) = 1 and (x, b) ~ Mo then x(t) < w(t) a.e. ; we omit the details. This work was carried out under the supervision of Dr. R. J. A u m a n n as a part of a doctoral thesis to be submitted at the Hebrew University.
REFERENCES 1. Aumann, R.J., July 1962, Markets with a continuum of traders I. Econometric Research Program, Research Memorandum No. 39. Princeton University. 2. Aumann, R.J. and Maschler, M., The bargaining set for cooperative games. To appear in No. 52 o f Annals of Mathematics Studies, Princeton University Press, Princeton, N.J. 3. Davis, M. 1961, Symmetric solutions to symmertic games with a continuum of players, Proceedings of the Princeton University conference on recent advances in game theory, held in October 1961. 4. Kalish, G.K., 1959, Generalized quota solutions of n-person games, Annals of Mathematics Studies No. 40, Princeton N.J. pp. 163-177.
1963]
QUOTA GAMES WITH A CONTINUUM OF PLAYERS
53
5. Maschler, M., December 1961, Stable payoff configurations for quota games, Econometric Research Program, Research Memorandum No. 36, Princeton University. 6. Peleg, B. On the bargaining set M0 of m-quota games, To appear in No. 52 of Annals of Mathematics Studies, Princeton University press, Princeton N.J. 7. Shapley, L.S. and Milnor, J.W., Feburary 1961, Values of large games, II: Oceanic Games, The Rand Corp., RM-2649, THE HEBREW UNIVERSITY OF JERUSALEM
ON O R D E R PRESERVING CONTRACTIONS* BY S. R. F O G U E L ABSTRACT
Let (f~,Y.,~) be a measure space and let P be an operator on L2(f~,Y.,p) with [IPI]< 1, Pf~ 0 a.e. whenever f > 0. If the subspace K is defined by
K--{x I
I[pnxll=ltP*nxll:-Ilxll
,
n--1,2,...)
then K = L2 (f~,~l,/~), where [1 c E and on K the operator P is "essentially" a measure preserving transformation. Thus the eigenvalues of P of modulus one, form a group under multiplication. This last result was proved by Rota for finite # here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. Introduction. The purpose o f this note is to generalize the results o f [2]. In [2] R o t a studies the eigenvalues o f modulus one o f a contraction P on L 1 (S, ~ , / 0 where # is a finite measure and P satisfies the following:
a. Pf > 0 w h e n e v e r f > 0. b. ess. sup. I Pf[ Z ess. sup. If[" This problem is related to the Frobenius Perron Theory. F o r bibliography on the subject we refer to [2]. O u r generalization is two-fold: 1. The measure # is n o t assumed to be finite. 2. The o p e r a t o r P is a contraction on L2(S,E,#) and is n o t assumed to be defined on LI(S, E, p). IfPhasnormonein t and Lo~ then, by the Riesz Convexity T h e o r e m it has n o r m 1 also over L2, thus 2 is weaker than R o t a ' s assumption. We shall use the m e t h o d o f the p r o o f o f T h e o r e m 2.2 and L e m m a 1.2 o f [1]. There the case #(S) < oo was studied. The results o f [2] are included in Theorems 1 and 3 o f this note. Let (S, E, p) be a measure space with/~ > 0. LEMMA 1. Let L be a closed subspace of L2(S,~,,l~), which satisfies: Received March 28, 1963. * The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force. 54
ON ORDER PRESERVING CONTRACTIONS
55
(1) l f f e L then R e f E L. (2) If f is real and f E L then Ifl eL. (3) If f > 0 a . e . , f E L , and c is a positive constant then min (f,c)E L.
Let E' contain all the sets, in E, whose characteristic functions are in L; then (a) The sets in Z' form afield; (b) The characteristic functions of sets in ~,' span L. Proof. Let f, g be real valued functions in L. Then max(f,g) =
k(lf- gl
+f+
g) e Z
min(f,g) = ½(f+g-[f-gl)EL. If tr and z are in E', let I(a) and I(z) denote their characteristic functions. Then max (I(tr), I(z))EL and min (I(tr), I(r))eL or: a U z E E ' and ~ n z E X ' . In order to prove (b) it is enough to show that the only function in L orthogonal to Z' is the zero function. Now if r e L is orthogonal to all functions I(tr),tr e E', then so is Ref. Thus we may assume that./is real. Let f + = ½( lfl + f ) E L and let c be a positive constant. Then, by (3), m i n ( f + , c ) e L and also f + - m i n ( f + , c )
geL. Let ~b = c- i rain (f+, c). Then h, = e- 1 min (5 q~,g) e L. Now: h~(to) = 0 iff÷(~o) < c, since then g(og) = 0, while: he(og) = 1 if f + > c + e. Also, for every to, 0 ~ he(co) ~ 1. Hence he(co) tends to the characteristic function of {~o > c) as ~ --} 0, thus I{co]f+(~o) > c} E Z and is orthogonal to f ; i.e., f < c a.e. for every c > 0. Therefore f + = 0 a.e. Applying the same argument to - f we g e t f = 0.
If+(o,)
REMARK. If p(f~)< OO then 1 E L 2 and Condition (3) is a consequence of Condition 2 and (3')
1 E L.
DEFINITION 1. An operator P on L2(S,X,/t) is called an order preserving contraction (O.P.C.) provided:
1. I f f e L 2 is real valued then so is Pf. 2. I f 0 - < t e L 2 a.e. then Pf>__ 0 a.e. 3. I f f E L 2 is real valued and f < ca.e. then P f < c and P ' f < c.
4. [[P[[ __<1. Note that the conditions (1) (2) and (4) are the same if we replace P by P*. Throughout this note P will always be an O.P.C.
56
s.R. FOGUEL LEMMA 2. I f P f = ei°fthen
PIfl =
[March
[fl.
Proof. We will need the inequality
IPFI<=PIFIa.e. Now if F is real this is immediate. Generally we have
]RePF I < P l F l a . e . since -4-ReF <=!FI. Also for every ;~ with [ ;~[ = 1
IRe,~PFI<=PI,~FI =
(*)
P I F I a.e.
Thus if IPFI> PIFI o n a s e t o f positive measure there is a set tr of positive measure, such that ifco~ tr then
IPF[>(I+6)PIFI,
l a r g P F - ~kl < ~
where 6 > 0 e > 0 and ~ can be chosen arbitrarily small. But then
Rel e-'~PFI > I PFI cos a > I FI
contradicting (*).
(This argument was suggested to us by Y. Katznelson). The proof of the Lemma is now straightforward:
llfll ~ >= II Plf111 [ISlI ~- (PISI,Ifl) >= I(ef,f)l
=
lifll ~
hence (PlSl, ISI~ = II Plsl II IlSll o~ Plsl =IslTHEOREM 1. Let L = {flPf =f} and let Z' contain all the sets tr in ~ such that I(a)¢ L. Then ~' is a field and its characteristic functions generate L, Proof. It is enough to verify Conditions (1), (2) and (3) of Lemma 1.The first condition is obviously satisfied. Now if lYl~ L then ISl ~ L by Lemma 2. Finally if 0 < f = Pf then P[min(f,c)] < Pf =f,
P[min(f,c)] < c ;
thus
P[min(f,c)] < min(f,c). Hence
P [ f - min(f,c)'] = f -
P[min(f,c)] > f -
min (f,c)
a n d f - rain (f, c) => O. We must have equality a.e., since l] P ][ =< 1,thus rain (f, c) e J[. DEF,N,TION
2. K
=
{st II ~sll-- II P*"Sil
:
lifll, n >__ ~}.
1963]
ON ORDER PRESERVING CONTRACTIONS
57
Now [iWf[[ = [[f[[ if and onlyif p , n p ~ f = f , and it is easy to check that P*nP~ is an O.P.C. Also l] P*~fi[ = ][f]] if and only if I~P*~f=f. Thus K is generated by characteristic functions of the intersections of the corresponding subfields of Y~. DEFINITION 3. Let E 1 contain all the sets, a, such that I ( a ) e K. By the above remarks %1 is a field and it generates K. THEOREM 2. The set K is a closed subspace of L2, invariant under P and P*. On K, P is a unitary operator. I f f _LK then weak lim F ' f = weak lim P * f = O.
Also, if tr e E 1, then PI(a) = I(z), where z e %I. Proof. It is enough to prove the last statement since the rest is proved in Theorem 1.1 of [11. Let a e %1 and PI(a) = f then 0 __
LE~IA 3. If f ,g are in K and are real valued, then P [min(f,g)] =min(Pf, Pg). If, in addition, f is bounded then P(f. g) = PfPg. Proof. Since P is order preserving P[(min(f, g)] __<min(Pf, Pg), and a similar relation holds for P*. Thus P*[min(Pf, Pg)] < min(f,g). Applying P to this inequality we get P [ m i n ( f , g ) ] < m i n ( P f , P g ) < P [ m i n ( f , g ) ] . In particular i f f and g are characteristic functions then
P(fg) = p f . Pg. The last part of the Lemma is proved by taking limits of sums of characteristic functions. THEOREM 3. I f Pf = eiOf then P [ (sgnf)2 [f[] = e2iO[(sgnf )2 If I]" Proof. It is well known that if P f = ei~" and I[ P II --< 1 then
llfllb. Plfl=lfl. Let s:--s(
P ' f = e-i° ((P*f,f) = e-i° Thus feK and, by Lemma 2, P I, = I~ by Theorem 1 ; thus
P [(sgnf) I~ if[ ] = PI, Pf = I~ e'°f = e'°I, (sgnf)
If l.
Then
58
s . R . FOGUEL
[March
On the other hand P [(sgnf) I, tfl ] = I, P [(sgnf)I,] P if[ = I~ If [P(l, sgnf) or
P(I, sgnf) = I~ P(I 8 sgnf) = ei°I, sgnf. Therefore P[(sgnf) 2 I~ If] ] = If] P[(sgnf) I. ]2 _-f e 2,oI~(sgnf) 2 Let e ~ 0 , then I, If]-~ Ill ; hence P [ ( s g n : ) 2 lYt] =
e2'°(sgnf)21:J"
Let us conclude with a uniqueness theorem. THEOREM 4. Let Pt and P2 be unitary order preserving operators. Then the subspace L = {J[ P l f = P2f} is generated by the characteristic functions contained in it. Proof. We will verify the three conditions of Lemma 1:
(1) If P l f = P2fthen Pl(Re f ) = R e P l f = Re P 2 f = P2(Re f). (2) I f f is real valued and P f f = PzJ then Pt(f+) = (Plf)+ = (P2/)+ = P2(f+) and P i ( f - ) = (Plf)- = (Pzf)- = P2(f-). Since Plf+ and PJ_ are positive, their sum is P~f and (Ptf+, Pif-) = (f+ ,f_) = 0. (3) It will be enough to show that Pi[min (f, c)] = min (Pif, c) f o r f > 0 and c a positive constant. Now P, [ min(f, c)] < min (P.,f,c) ; thus for P~* we have P*[ min (P,f, c)] < min (f, c). Applying Pi we get min (Ptf, c) £ [P. rain (f, c)]. Thus, in order to find whether two unitary order preserving operators are equal, it is enough to show that P l f = P2f, whenever f is a function such that the functions I{to If(to) ~ A} generate L2(S, X, p).
1963]
ON O R D E R PRESERVING C O N T R A C T I O N S
59
BIBLIOGRAPHY 1. Foguel, S.R., Powers of a contraction in Hilbert space, To be published in the Pacific J. of Math. 2. Rota, G.C., 1961, On the eigenvalues of positive operators, Bull. Amer. Math. Soc., 67, 556-558. THE HEBREW UNIVERSITYOF JERUSALEM
