Israel Journal of Mathematics, Volume 3, Issue 3 (1965)

THE COMPOUND DECISION PROBLEM IN THE OPPONENT CASE BY

ESTER SAMUEL ABSTRACT

The compound decision problem is considered under the assumption that Nature plays the role of an opponent. The empirical Bayes approach is a particular case of this approach. It is shown that in many instances sequential compound rules with good asymptotic properties for the fixed caw have similar good properties in the opponent case.

1. Introduction and Snmmary. Consider a general statistical decision problem, with its usual components: The parameter space fl with elements 0, the action space 9.[ with elements A and the loss function L(A, 0) ~ 0 defined on 9~ x fl. The action chosen depends upon the value x e ~r of an observable random variable X which is distributed according to the known distribution Po, when the true parameter is 0. A decision function ~b is a function measurable in x, which for each fixed x is a distribution over (% a~), (where ore denotes a suitable o-field of subsets of B). The risk function R(~, 0) of ~ is the expected loss incurred by use of 4, considered as a function of 0 e[l. In the Bayesian approach 0 is considered as a realization of a random variable A with (a priori) distribution G over (fl, on). The Bayes risk of ~b with respect to G is R(~b, G) = faR(J, O)dG (0).

(1)

(1) is minimized (for fixed G) by any Bayes rule (PGfor which infR(~, G) = R(::~, G) = R(G).

(2)

R(G) is called the Bayes envelope functional. We assume that there always exists a minimizing ~ .

The sequential compound problem has been of interest lately. (See [3], [9], [4], [11] and [12]). Such a problem arises when one is confronted with the same statistical decision problem, called the component problem, not only once, but Received Feb. 17, 1965 117

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sequentially n times. One therefore has a vector 0n = (01, .",On) of (unknown) parameters, 0~el), and a corresponding vector Xn = ( X I , ' " , X , ) of random variables, where the Xi's are assumed independent, and Xi has distribution Is,. In this situation one can use a sequential compound decision rule 0n = (tPt, "", $,) where $i depends on X~, and for each x~ is a distribution over (9.I, o'~) by means of which the i-th action is chosen, i = 1,..., n. (Generally $i may depend on previous actions, i.e., there may be dependence between the randomizations.) We denote the random loss incurred on the i-th decision by L($i, 03, and its average over the n decisions by L(~n,0n). We shall define this average to be the compound loss. The corresponding expected values are R($i, 03 and R(dpn,0n) respectively, the latter being the compound risk of qbn. Let ~ = (~b1, ~b2, ...) be a strongly sequential compound rule (i.e. a sequene where ~bi is as described above but does not depend on n), and denote by ~n its initial n-vector. Since we shall be interested in asymptotic properties of sequential compound rules, we shall for convenience limit our attention to strongly sequential rules. Let 0 = (01,02, ...), 0i e f~, be a sequence of parameter values, with 0n as initial n-vector. Let G, denote the empirical distribution of 0n, i.e. the distribution over (f~, an) which assigns to 0 the probability s/n if 0i = 0 for s values of i, i = 1,..., n. If Gn were known in advance, one could, by using for every i a rule ~b~ for the component problem with the observation x~ to decide on 01, achieve R(Gn) defined in (2), as the value of the compound risk at every On which has Gn as its empirical distribution. Gn is, however, usually unknown, so this simple rule cannot be used. One is therefore led to consider rules #p with good asymptotic properties, in the following sense: dp is said to have property (A) if for every 0 = (01,02,..') and every t > 0 there exists a N(0, 5) such that for all n > N(0, 5) R(d#,, 0n) - R(Gn) < ~.

(A)

If N(0, e) can be chosen to be independent of 0, d~ is said to have uniform property

(A). is said to have property (B), almost everywhere if (B)

lira sup [L(On, 0n) - R(Gn)] < 0

a.e.

n--* CO

is said to have property (B) in probability if for e > 0 (B)

lim P(L(tb,, On) - R(G,) < 8) = 1. n--~ oo

For the problem of testing a simple hypothesis against a simple alternative rules having uniform property (A) are exhibited in [9]. In [11] they are shown to have

1965] THE COMPOUND DECISION PROBLEM IN THE OPPONENT CASE

119

property (B). For some estimation problems, rules with properties (A) or (B) are exhibited in [12]. In the present paper we change the theoretical setup, and consider what we call the opponent case. Rather than considering a fixed but unknown sequence 0 we consider Nature as an active opponent of the statistician. We assume that Nature has a strategy ~ by means of which the parameters (01,02"") are chosen. Let A l denote the random variable whose observed value 0i determines the distribution Po, of Xi. We shall let Ai=Ai(y)=A~(t~,Xi_l, Ai_l,Zs_l). Here Zj denotes the auxiliary random variable used to select the j-th action (needed when the j-th decision is randomized). Thus we let A i depend upon all previous random variables, and we may even let it depend upon the functional form of tb. Clearly under these assumptions the X~'s are no longer independent. The conditional distribution of X~, given A~ = Oi, is still/'0,. We shall call ~ a strategy only if it together with ~ and the family P0, 0~ ~, generates a probability measure on the measurable space of interest. This probability measure will be denoted by Pr, expectation with respect to it by Er, and we assume that all conditional expectations considered in the sequel arc meaningful. We shall maintain the notation introduced earlier, but replace 0i by the random As, and we let Kn denote the (random) empirical distribution of A,. Corresponding to the earlier definitions we say that ~ has property (Ao) if for every strategy ~ and every e > 0 there exists N(7, e) such that for all n > N(?, e) (Ao)

E~[L(d~,An) - R(K.)]
0 (Bo)

lim Pr(L(~,,A,) - R(K,) < e) = 1. n--* oo

Clearly (Ao) and (Bo) are stronger properties than (A) and (B), respectively, since (A) and (B) follow from (Ao) and (Bo) if we consider only such strategies which with probability one select a specified sequence 0. Notice that whenever (5) below holds, property (Bo) implies property (Ao). One may object to the consideration of the opponent case in statistics, since it seems unrealistic to believe that Nature chooses the parameter values dependent upon the previous decisions of the statistician. From this point of view the opponent case seems more justified in a game-theoretic setup. It has been dealt with for nonstatistieal games in [1] and [2] by Blackwell, who exihibits rules having

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a property corresponding to (B0). We remark that a similar criticism is equally valid regarding the minimax attitude, which nevertheless has received much attention in statistics. Our aim in the present paper is to show that the rules considered earlier for the sequential compound situation actually have properties (Ao) or (Bo), respectively. One particular example of the opponent case is the empirical Bayes approach. This approach is certainly motivated in statistics. (See e.g. [7], [8] and [10].) In the next Section we proceed to prove that every ~p with property (A) is an asymptotically optimal empirical Bayes rule. In the Section thereafter we show that an optimal strategy for the opponent is the repeated maximin strategy of the component problem. In Section 4 some theoretical results are stated, which are used in Sections 5 and 6 when dealing with the problem of testing simple hypotheses, and the estimation problem, respectively. 2. The empirical Bayes approach. In this approach it is assumed that A s are independent, identically distributed random variables, with unknown apriori distribution G over (~,(ra). Let G n denote the product distribution over the n-fold Cartesian product (~", tTu.), generated by G. Corresponding to (1), the Bayes risk on the n-th decision will be denoted R(~,, G~), and is the expectation with respect to G ~, of R(~b,,A,). ~b is called an asymptotically optimal empirical Bayes rule if for every (apriori) distribution G lim R((p,, G") = R(G).

(3)

n--~ O0

(3)

implies lim R(~., G") = R(G).

(4)

n--+ OO

It follows from [13], Theorem 1 that always R(c~,, G")> R(G), and hence (4) implies (3), and asymptotic optimality could be defined either by (3) or (4). It is well known that there exist component problems for which (3) cannot be achieved. See [10] and [8]. We shall need the following lemma, which is quite well-known. LEMMA 1. The functional R(G) is concave.

Proof. Let 0 =

__pR(dp~, GI) + qR(dp~,, G2) = pR(GI) + qR(G2). TIIEOREM I. If

(5)

sup sup L(A, 0) = M < oo Ae~IOen

then every ¢~with property (A) is an asymptotically optimal empirical Bayes rule.

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THE COMPOUND DECISION PROBLEM IN THE OPPONENT CASE

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Proof. Let ~G denote the strategy by which the At are independent, identically distributed according to G. Then for every real t Err K.(t) = G(t), and it follows by Lemma 1 and by use of the usual approximation by step functions that

ErGR(Kn) 0 be given. If R(t/) is differentiable then by Lemma 1 of [9] there exists a 6 > 0 such that for any versions of (10) (12)

max 0=0,1

IR(q,o) -

R(t¢, 0) l < e/3 for

[ r / - r/* [ < 6.

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ESTER SAMUEL

[September

Let co denote a generic point in our measure space and let

Let NI be a fixed integer greater than 2/6 + 1. For i > NI we have by Chebychev's inequality (13)

P , [ [ P , - x - 4,[ ->__6] Z P~[IP,-1 - '~,-11 >- 6 - i -1] =< V / [ ( i - 1),~z - 26]

whero V =. max{Eoh(X) 2, EI[h(X) - 1]z}. With our definitions we have (14)

R(t* ] ~-,_1) = R(t°,_t ,A,)

where t~o is defined through (10) with the arbitrary part taken to be 0. Thus for co ~ Sl- 1 we have from (12) and (14)

(15)

JR(t71

-

I < 5/3

and hence for i > NI, it follows from (13) and (15) that (16)

E,/.(t?, Ai) = E,R(t* I ~ ,_,) < E,R(t~,,A,) + , / 3 + M V/6[(i - 1)6 - 2].

Now for n ~_ 3 M N 1 / s n - 1 ~N~ 1 L(t*, Ai) < e/3 a.e. Pr and for n > N2 sufficiently large n - t ~E~ffiN.+ 1 [(i -- 1)6 -- 2]- 1 < e6 ]3MV. Thus for n > max(N2, 3 M N I / e ) we have from (16) g , L ( t * ~ , a . ) - e , n -1 ~ L ( t L A . ) < E ~ n - i ~. R(ta,,ai ) + ~, i=1

iffil

and uniform property (Ao) for t* follows from Lemma 2. We shall show that (9) holds a.e. Pr for t*. This follows immediately from (11), (12) and (14), since here R(~^,,A~) = R(ta,,Ai). Thus t* has property (Bo) a.e. Pr, and the proof is complete. We remark that uniform property (A) and property (B) for t* was proved under the same condition on R(~), as in Theorem 2 above. The rule "/is randomized. Let Pi = Pi(Xi) be defined as before, but with h(x) a bounded estimator. For i = 1,2,..- let Z+=(zil,z~2) be independent random variables, uniformly distributed on the unit square, Z~ independent of X+,AI. For i ffi 2, 3,... let mi-x =~ rn(Zt, xi_l) = Pi-x(x~-x) + i -1/+

Zt2

1 + i-1/'*(z+1 + z+2) and define mo arbitrarily. ! is defined by means of ?i(x+) = t,,,_, o (xi), i = 1,2,..., where the right hand side is defined through the nonrandomized version of (10) mentioned earlier. We state without proof the following

1965] THE COMPOUND DECISION PROBLEM IN THE OPPONENT CASE

125

THEOREM 3. For any two distributions Po,P1, t has uniform property (Ao) and has property (Bo) in probability. In connection with Theorem 2 and 3 we remark that, as has been pointed out to the author by Professor H. Robbins, the third paragraph on p. 1084 of [9] is incorrect, in that there the opponent case sould have been considered. 6. The estimation problem. Rather than discussing the opponent-case properties of all the compound estimators considered in [12] in full generality, we shall consider a particular example. Let the component problem be that of estimating the parameter 0 of a Poisson distribution, where 0eg2 = {0:0 < ~ < 0 < fl < oo} (where ~,fl are otherwise arbitrary). Let 91 = f2, L(A, 8)= ( A - 8) 2, so (5) holds with M = ( f l - ~t)2 . We shall consider nonrandomized estimators only, and let ~b(x) denote the value in 9I chosen with probability one. Let ]¢j(x) be 1 or 0 according as Xj = x or X 1 ve x, and for x = 0,1,... let |

(17)

q i ( x ) = i -1

t

Z Y~(x), q~:,(x)=i -~ Z P(Xj=xIA,).

j=l

j=l

The sequential compound rule ~ considered in [12] (see also [7]) is defined by (I8)

dpi(xi) = (xi + 1)qi-l(xi + 1)/qt-l(xl), ~, or fl

according as the first term on the right hand side of (18) is between 0t and fl, less than 0t, or greater than fl, respectively. We shall prove THEO~M 4. The rule ~ defined by (18) has property (Bo) a.e. P r Proof. We shall show that (9) holds a.e. P r It is easily checked that (cf. e.g. [7], (18) or [12], Section 5.) (19)

q~r,(xi) = (x, + 1)qr, (x i + 1)/qr,(x~)

and ct __ 0 there exists a finite N~ such that Pe(X 1 - ~/2M, for all 0 e ft. Thus (20) holds uniformly in 0 =< x =< N,, i.e. for every 6 > 0 there exists 1(8,6) such that

126

ESTER SAMUEL

(21)

Pr(lqbK,(x) - q~(X,_ l, x) I < e / 6 f l f o r all i > I(e, a n d all 0 < x < N~) > 1 - 6.

But s t r a i g h t f o r w a r d a p p r o x i m a t i o n s show t h a t (21) implies P~(IR(q 5, I ~ ' , - i) - R((°r,,A,)I < ~ for all i > 1(5, 6)) > 1 - 6, which is j u s t an explicit f o r m o f (9).

REFERENCES 1. D. Blackwell, Controlled random walks, Prec. International Congress Math. 3, Amsterdam (1954), 336-338. 2. D. Blackwell, An analogue of the minimax theorem for vector payoffs, Pacific J. Math. 6, (1956), 1-8. 3. J. F. Hannan, The dynamic statistical decision problem when the component problem involves a finite number, m, of distributions. Abstract, Ann. Math. Statist, 27 (1956), 212. 4. M. V. Johns Jr., The parametric and nonparametric compound decision problem in the sequence case. Abstract, Ann. Math. Statist. 34 (1963), 1620-1621. 5. M. Lo~ve, Probability Theory. 2nd ed. Van Nostrand Comp., (1960). 6. H. Robbins, Asymptotic subminimax solutions of compound statistical decision problems, Proc. Second Berkeley Symp. Math. Statist and Prob. Univ. of California Press, (1951), 131-148. 7. H. Robbins, An empirical Bayes approach to statistics, Prec. Third Berkeley Syrup. Math. Statist. and Prob. 1, Univ. of California Press. (1955), 157-163. 8. H. Robbins, The empirical Bayes approach to statistical decision problems, Ann. Math. Statist. 35 (1964), 1-20. 9. E. Samuel, Asymptotic solutions of the sequential compound decision problem, Ann. Math. Statist. 34 (1963), 1079-1094. 10. E. Samuel, An empirical Bayes approach to the testing of certain parametric hypotheses, Ann. Math. Statist. 34 (1963), 1370-1385. 11. E. Samuel, Convergence of the losses of certain decision rule~for the sequential compound decision problem, Ann. Math. Statist. 35 (1964), 1606-1621. 12. E. Samuel, Sequential compound estimators, Ann. Math. Statist. 36 (1965), 879-889. 13. E. Samuel, On simple rules for the compound decision problem, J. Roy. Statist. Sec. (]3) 27 (1965), 238-244. 14. E. Samuel, On the sequential compound decision problems in the fixed case and the opponent case. Unpublished (1964). THE HEBREW UNIVERSITY OF JERUSALEM

THE KERNEL OF THE COMPOSITION OF CHARACTERISTIC FUNCTION GAMES(1) BY

BEZALEL PELEG ABSTRACT

The structure of the kernel of a composition of two games is investigated; a comparison with the results for Von-Neumann and Morgenstern solutions is included. 1. Introduction. In this paper we investigate the kernel of the composition, in the sense of yon N e u m a n n and Morgenstern, of two games. Sections 2, 3, and 4 contain the necessary definitions, preparatory lemmas and an example which shows that the phenomenon of transfer occurs in the kernel. In Section 5 we give a complete description of the kernel of a composition and obtain bounds on the transfer. The case of composition of two ordinary three-person games is studied in full detail in Section 6. In Section 7 we apply the results of Section 5 to constant-sum games; it turns out that the results obtained are similar to the results of yon N e u m a n n and Morgenstern for solutions. 2. Definitions. Let N = {1, 2, . . , n} be a set with n members. A characteristic function is a non-negative real function v defined on the subsets of N which satisfies

(2.1)

v(~b) = 0, and v({i}) = 0, for i = 1, ..., n.

The pair (N; v) is an n-person game. The members of N are called players. Subsets of N are called coalitions. The game (N; v) is an ordinary game if (2.2)

v(S) < v(N),

for all

S c N.

Let G = (N; v) be an n-person game. An individually rational payoff vector (i.r.p.v.) is an n-tuple x = (xl, "", x,) of real numbers which satisfies: Received June 21, 1965 (1) The research described in this paper was partially supported by the U.S. Office of Naval Research under Contract Number N62558-4355. 127

128 (2.3)

BEZALEL PELEG xi > O, i = 1, ..., n

[September

(individual rationality),

and (2.4) i=l

x i = v(N).

The set of all the i.r.p.v's is denoted by A(G). Let i a n d j be two different players. We denote by J i j the set of all the coalitions which contain player i but do not contain player j; i.e., (2.5)

J't~={S: ScN,

ieS

and j ¢ S } .

Let x be an i.r.p.v, and let S be an arbitrary coalition. The excess of S with respect to x is(z) ; (2.6)

e(S, x ) = v(S) - ~ xi

The maximum surplus of i over j with respect to x is (2.7)

si~(x) = max {e(R, x) : R e,Y'o}

i is said to outweigh j with respect to x if (2.8)

s~(x) > s~(x)

and

xj > 0.

x is balanced if there exists no pair of players h and k such that h outweighs k. The kernel(3), (4)of G,~Y'(G), is the set of all balanced i.r.p.v's. Let c be a real number which satisfies the inequality c < v(N). We define a new game G(c) = (N; vc), with the same set of players N as in G, whose characteristic function vc is defined by

(2.9)

Oc(S)

[ v(S),

s ~N

1 (v(N)-c,

S=N.

The core (s) of the game G is the set of all i.r.p.v.'s x which satisfy (2.10)

e(S, x) < O, for all S = N.

The strict core is the set of all i.r.p.v.'s which satisfy (2.11)

e(S, x) < 0, for S = N, S ¢ ~ and S ¢ N.

(2) e(~, x) is taken as equal to zero. (3) The reader is referred to 12] for a comprehensive introduction to the kernel theory, and to [4] for recent developments. (4) For the coalition structure (N} (see 12]), (5) See [7, page 11 for the origin of this concept.

1965l

THE COMPOSITION OF FUNCTION GAMES

129

3. The kernel of the composition of two games. Let Gt-----(Nt; vt) and G2 = (N2;v2) be two games with disjoint sets of players, (i.e., N t N N2 =J25). The composition(6) of G t and G2 is the game G = (N; v) which satisfies (3.1)

N = Nt k) N2,

and (3.2)

v(S) = vt(S N Nt) + v2(S N N2), for all S c N .

EXAMPLE 3.1. Let G1 and G2 be two three-person constant-sum games in (0, 1) normalization, with disjoint sets of players. In G, the composition of Gt and G2, each pair of players of GI, or of G2, are symmetric. Hence, by [4] Theorem 9.3, the i.r.p.v's of the kernel of G are of the form (x,x,x,2/3 - x , 2 / 3 - x , 2 / 3 - x ) , where x is limited by the requirement m a x ( 1 - 2 x , 2 / 3 ) = max(2x-1/3,2/3), i.e., 1/6<x hi(x). It follows that there exists a coalition S which satisfies i e S and g~(x) = e(S, x) > hi(x). Let j ~ N - S. sij(x) > e(S, x) > hi(x) > sj~(x). Since x is balanced, x~ = 0. Hence

e(S,x)=v(S)-

~, X k = v ( S ) - k~S

Z Xk < v ( N ) k~N

~, Xk = O . k~N

Thus g~(x) < O. But we have assumed that g~(x) > h~(x) > O. Hence, our assumption cannot hold, and the proof is complete (6) See [5], Chapter IX, for the origin of this definition.

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BEZALEL PELEG

[September

The Second lemma applies to modified games. LEMMA4.2. Let x ~o,~F(G(c)) and i ~ N. I f f (x) > 0 and x i > 0 then gi(x) ~ hi(x). Proof. f ( x ) < max(g~(x), hi(x)). Assume hi(x ) > gi(x); then f ( x ) ~_ h~(x). Our assumption f ( x ) >- 0 implies that f ( x ) > hi(x). Thus f ( x ) = h~(x). It follows that h~(x) = e(S,x) for a coalition S which satisfies S # ~ and i ¢ S . Let j e S'sji(x) > e(S, x) > gi(x) > sij(x). Thus j outweighs i, which is impossible. Hence our assumption hi(x)> gi(x) cannot hold. 5. The Structure of the kernel of a composition of two games. Let G 1 ----(N 1; vl) and G2 = (N2, v2) be two ordinary games with disjoint sets of players, and let G = (N; v) be the composition of G 1 and G2. An i.r.p.v, z ~ A(G) is a real function defined on N which satisfies (2.3) and (2.4). The restrictions of z to Nx and to N2 will be denoted by x and y respectively. The amount of transfer exhibited by z is given by

(5.1)

t = t(z) = vl(Nt) -

~, x~ i~Nj

The games Gl(t ) and G2(-t) (see (2.9)) are related in a natural way to z. Clearly x e A(GI(t)) and y ~ A(G2(- t)). In denoting function of x and y we adopt the following convention: functions depending only on x[y] will bear the superscript " 1 " [ " 2 " ] . Thus, e.g., if i , j ~ N l then s:j(x) is the maximum surplus of i over j with trespect to x (see (2.7)) in the game Gl(t). LEMMA 5.1. Let z ~ A ( G ) and i, j e N l [ i , j e N 2 ] , i outweighs j with respect to z (see (2.8)) in G, if and only ifi outweighsj with respect to x[y] in Gt(t ) [ G 2 ( - t)], where t is defined by (5.1). Proof. Clearly it is sufficient to consider only the case i,j ~ Nt.

su(z ) = max{e(S, z) : S ~ Y',j} = max{e(S, z) : S n NI e ~'ij} = max{e(S, z) + e(R, z) : S ~ NI, SEY-ij and R c N2} = max{el(S, x) : S c NI, S e ~iy} + max{e(R, z) :R c N2}

= s~(x) + max{e(R, z) : R c N2} Similarly sj,(z) = sJ,(x) + max{e(R, z) :R c N2}. Hence s~(x) > s~,(x) if and only if slj(z) > sji(z). Since Xy = z j, the proof follows. COROLLARY 5.2. I f Z E.)[" (G) then x ~ )F ( G l ( t) ) and y e :,'f" (G2(- t)).

LEMMA 5.3. I f z ~ A(G), i ~ N 1 and j ~ Nz then

sij(z ) = g~(x) + h~(y)

(Sy,(Z)= g~(y) + h](x)).

1965]

THE COMPOSITION OF FUNCTION GAMES

131

Proof. Again only the first half of the lemma will be proved.

sil(z) = max{e(S, z) : S ~ }

= max{e(S, z) + e (R, z) : i ~ S, S c N I ,

and R ~ N21j ~ R} = max{e(S, z) : i ~ S, S ~ N~} + max{e(R, z) :j ~ R, R c N2}

= g~(x) + h~(y), (see (4.2) and (4.3)). We remark that g~(x) [hff(y)] is computed here with respect to Gl(t) l-G2(-t)]. THEOREM 5.4. ~ ( G ) n rA(G1) x A(G2) ] = ~3~"(GI) x ~ ( G 2 ) . Proof. Corollary 5.2 implies that -,~(Gl) x X'(G2) ~ oE'(G) ~ rA(Gt) x A(G2)]. Let now x ~ J~(Gt), y E)E'(G2) and z = (x, y). If i and j both belong to Nt, or both belong to N2, then, by lemma 5.1 j does not outweigh i with respect to the game G. So let MeN 1 with xi > 0 and j ~ N 2. By L e m m a 5.3 so(z ) = g~(x) + hi(y) and sj,(z) = g](y) + h~t(x).We shall show that g~(x) > hi(x). To prove this we observe that if f l ( x ) < 0, (see (4.1)) then gtl(x ) = h~(x) = O, and i f f t(x) > 0, then by Lemma 4.2, g~(x) > h~(x). By I_emma 4.1 h2(y) > g~(y). Thus so(z ) > sji(z ) and j does not outweigh i. Clearly we can interchange N t and N2 in the above argument. Thus the proof is complete. The following example shows that our assumption that both G t and G2 are ordinary games is essential for the validity of Theorem 5.4. But since constantsum(7), superadditive(7), or even monotonic(7) games are ordinary games, (2.2) is not too restricting assumption. EXAMPLE 5.5. Let us consider the following two games: ({1,2};u1) , where ul({1, 2}) = 1, and ({a, b, c}; u2), where u2({a , b})= 1 + e , 0 < e n

s(u).

Since • is convex function satisfying the growth condition ¢ ( 2 u ) < K ¢ ( u ) for u > 0 there exists a positive number 2 such that ~(~ u) < 2¢(u) for u > 0. Hence the inequality above implies S(U)> 0. Since (X, S,p) is an infinite non-atomic measure space there exist 2"-1 pairwise disjoint measurable sets {Ai}2=~ 1 such that

p(Ai)- 2 . _ 1nS ( U ) • Let tSAm1 i j .n, = l , for 1 - < i < 2 " - 1 , be a measurable partition of A~ such that 1

p(AT) - 2._ 1 S(U) Now we define n functions {f~}i%1 in L* by setting 2n-1

fi = Z K=I

~ CIT~UK~At K. t=l

It is verified that

M@ f' + f2"4-'" +--f" ) n

nS,(U;~) = 2,_aS(U)

> 1

for any combination of signs while M(f,) = 1 for 1 _< i < n. Thus a contradiction on the choice of ~ is obtained and the p r o o f of the Theorem is complete. We proceed to the case when 0 < p(X) < oo. We state first a definition and establish some auxiliary lemmas.

142

KONDAGUNTA

SUNDARESAN

[September

Following the terminology in Nakano 13] the Banach space L* is said to be uniformly finite if sup

M ( K f ) < oo

M(f)< 1

for any positive real number K. LEMMA 1. If L* is uniformly finite then if (f,},_>_i is a sequence of functions intheunit ballofL* such that 1 a s n ~ oo then M ( f , ) ~ 1 as n ~ oo. The proof of the lemma is an immediate consequence of Th. 4 on p. 224 in Nakano [-3].

IIf, I1-

LEMMA 2. I f L* is uniformly finite and if there exist real numbers t and tl 0 < t, tl < 1, such that if (f~},"=1 are any n functions in the unit ball of L* with M(fi) >__1 - t for 1 < i < n then for some choice of signs M ( fx + ' " + - f "

) < 1 -

L* is uniformly non-l~ 1) . Proof. If L* is not uniformly non-1,(t), then for any sequence of reals ~i such that ~j > 1 and ~j ~ 1 there exist n sequences {fiJ}j____1, 1 < i < n in the unit ball of L* with the property 1 --

f / +- "" +-f~


C > O a n d q~(vo)if ~(Vo) -

C ~ U > 0 for l < i < n .

#(X) "

ProoL Let us assume that L* is uniformly non-l~(1). Then definition 2 guarantees the existence of real number Go > 1 such that if {f~}~ 1 are any n functions in the unit ball of L* then for some choice of signs M

o

n

We shall prove that ~ fulfills the condition (i) with the choice of ~ = Go. Assuming n = 1 such that the contrary there exist non-negative numbers {U~}~ n

2"- 1

S(U) > /z(X---~ and SI(U;~) > ~ n Since

S(U).

n

~(x) >= s(u) there exist 2"-1 pairwise disjoint measurable sets tYAij~ = 1l such that ~2,n

/~(A~) = 2,_1S(U) for 1 < i < 2"-1

144

KONDAGUNTA SUNDARESAN

[September

Now we can complete the proof by constructing the functions {fi}7= i as in theorem 1 contradicting the choice of ~o. Next we shall show that

di)( v° n ¥ ) # !,(Vo). If possible let Vo) +

n

= 1¢(%).

Since*(v o) =

n

n

and X is a nonatomic measure space there exist n pairwise disjoint measurable sets {~}i=lA ~ such that da(Vo)l~(Ai) = 1. Let f i = vozA, for 1 _< i _< n. Then clearly M(f~) = I[fi I[ = 1 for 1 -< i < n and for all choices of signs

since

/')0 by Remark 1. Hence a contradiction arises on the choices of Go. Next we shall prove that if • satisfies the inequalities (i) and (ii) then L* is uniformly non-l~ 1). If • satisfies inequality (i) and ~(v~) = C then by choosing the vector U such that CiU = u > Vl it is verified that ~(¢u) < 2 " - l ~ ( u ) for all u > Vl. Thus L* is uniformly finite by Lemma 3. Since Vo)

* n

#n

1 ¢(0o)

and • is continuous it follows by Remark 1 that there exists a real number vl such that 0 < vl < Vo, ~(v~) > 0, and # nl ~(01) # ~1 ~(Oo). Let~(vl) = 0~(Vo) = 0--/,(X) where 0 < 0 < 1. In the condition (i) we can assume that C > n [#(X). Let K = U I U e R" and/~(.~) < S ( U ) < C . We note that S I ( U ; 1) 1 such that

UNIFORMLY NON-1(1) ORLICZ SPACES

19651

2"-- 1

S~(U;~I) < ~

n

S(U)

145

U~K.

forall

Let Go = min(~l,¢). Then for all U such that

S(U) >_

0/7 ,

n-1

SI(U;¢o)< 2 n S(U).

Let t be a real number such that 1 - t > O. We shall prove that there exists an q, 1 > r/> 0 such that if {f~ }~'=t are n functions in the unit ball of L~ with M(f3>l-t

for l < i < n

then for some choice of signs

M ( fj+f2+'''+-f~n Let E = {x I I2~'=1 ~(f,(x)) > ~M

( fl+f2+'"n

)
0 and CiU = 0 for 2 < i < n it is verified that (A) for u > 0,

o ( u ~) ~=l @(u). Similarly if n-1

SI(U ; ~) C > 0 then there exists a v > 0 such that (B) for all

u>=v>O,

() If • is the Young's complement of O, and if • satisfies (.4) or (B) then there exists a constant K > 0 such that ~(2u) < K~(u) for all u > 0 or ~(2u) < K~(u) for large values of u according as (A) or (B) is true. T~OREM 3. IlL* is uniformly non-!~1) then it is reflexive. Proof. Since • and • satisfy the growth conditions ensuring that/.,0 and L , are linear it follows from theorems 4 and 5, Rao [4] that L~ is reflexive.

REFERENCES 1. James, Robert C., Uniformly Non-square Bannch Spaces, Ann. Math. 3, Vol. 80 0964), 542-550. 2. Luxemburg, W., Banach Function Spaces, Thesis., Technische Hoge School te Delft, 1955. 3. Nakano, H., Topology and Linear Topological Spaces, Maruzin and Co., Tokyo, 1951. 4. Rao, M. M., Linear Functionals on Orlicz Spaces, Nieuw Archief Voor Wiskunde 3, XI (1964), 77-98. 5. Weiss, G., A Note on Orlicz Spaces, Portugaliae Math. 15 (1956) 35-47. 6. Zaanen, A. C., Linear Analysis, North Holland Publishing Co., Amsterdam, 1956. CARNEGIE INSTITUTEOF TECHNOLOGY PITTSBURGH, PA, U.S.A.

ON FINITE DIMENSIONAL SUBSPACES OF BANACH SPACES BY

A. LAZAR A N D M. ZIPPIN* ABSTRACT

The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces. 1. It is well-known (by the Hahn-Banach theorem) that for each element x in a Banach space X one can find a functional feX* such that Ilfll--1 and x ll. The following natural question arises: Given a finite dimensional subspace E c X, is it possible to find a finite dimensional subspace F c X* such

f(x)--II

that for each x e E IIx II -- sup,, s~ I f ( x ) I ? In this paper we show that the answer is negative, and investigate some similar properties concerning finite dimensional subspaces. We prove that the spaces co(S) and real Lp, where p is an even integer, satisfy the above-mentioned condition, while C and Lp for all other p's do not satisfy it. The terminology and notations are generally the same as in [1]. Sx denotes the closed unit ball of the Banach space X. If F is a subspace of a conjugate space X* then F.L = {x:xeX,f(x) = 0 for all f~F}. If E c X then E ~-= {f:f~X*, f(x) = 0 for all x e E}. 2. Let us discuss the following conditions on an infinite dimensional normed space X, concerning its finite dimensional subspaces: (1) For every finite dimensional subspace E c X there exists a subspace F = X* such that for each x e E 1[x II -- supy~ sF If(x) l and F_L is infinite dimensional. (2) For every finite dimensional subspace E c X it is possible to find a finite dimensional F ~ X* such that for each xEE llxll= supsos lf(x)l. (3) For every finite dimensional subspace E ,-- X there exist a finite dimensional subspace G = X such that E c G and a projection PG of X onto G with II II = 1 Denote by ~¢~ the class of all normed spaces satisfying condition (i) i = 1,2,3. It is obvious that if we require in (1) that F is w*-closed ~¢1 will not change. Of course, d l -~ ~ 2 ~ ~¢3. In §4, 7 we shall show that ~¢1 ~ ~¢2 ~ ~¢a. The following two lemmas give equivalent conditions to (1) and (2) respectively. Received July 15, 1965. * The contribution to the paper of the second author is part of his Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor A. Dvoretzky. The authors wish to thank Professor A. Dvoretzky for the interest he showed in the paper and for his helpful advice. 147

148

A. LAZAR AND M. ZIPPIN

[September

LEMMA 1. Let X be a normed space; then X e ~ 1 if and only if for every finite dimensional subspace E ~ X there exists an infinite dimensional closed subspace G c X satisfying the following conditions: (a) G n E = {0}. (b) If H is the subspace spanned by the elements of E and G, then there exists

a projection Plz of H onto E along G.

(c) IIe~ll : J. Proof. Necessity: Take for G the subspace F L. If x e G O E then Ilxll=supsos~lf(x)l=0, hence x = 0 ; if x = e + y , eeE, y ~ G then I l e + y i l > s u p s ~ s ~ l f ( e + y ) l = s u p f ~ s ~ l f ( e ) l = l l e l I . Hence, the transformation P~ defined by Pr.(e + y) = e is a projection of H onto E along G and since IIe II --< II e + Y II, II P~ II -- 1 Sufficiency: For every f e E* define f(e + y ) = f(e) for each e + y e H. Now, f e H*, and f l y ) = 0 for every y e G. By (b) and (c) II/ll~ = suPlle+,ll=l f(e + y) = SUPllell__= 1 (n = 0) and the other are positive but less than 1. Hence, the smallest closed subspace of l~ on the unit ball of which every vector from our plane attains its norm is It itself. The same plane taken this time in the space m shows that this space also has not the property (1). (We use the fact that 1 is w*-dense in m*). Considering in C[0,1] the subspace spanned by f l ( x ) = x, f2(x) = 1 - x 2 it can be proved that C[0,1] does not satisfy condition (1). Now we are able to display a space which belongs to M1 but fails to belong to M2. If X is a space which does not belong to ~¢2, the space Y= (/2 ~ X ) t~ is the desired one. Indeed, the conjugate of Yis (/2 • X*)m and if E is a finite dimensional subspace of Y any element of E attains its norm on the unit ball of (P(E) ~ X*)m c (12 ~ X*)m, where P denotes the natural projection of Y onto 12. The annihilator of P ( E ) ~ ) X * is the orthogonal complement of P(E) in 12 which is obviously infinite dimensional. But Y fails to have the property (2) since its subspace X has not this property. (See Remark 3). If we suppose that X does not belong even to . d t, like the space c for instance, we see from the above example that property (1) of a space is not inherited by its subspaces. 5. We shall prove now that the property (1) is not valid in 11. Thus no infinite dimensional L-space has property (2). Let a = {at} ~ 1, b = {bt}~-°=1 be two elements of l~ such that

at a~+~ b--~> bt+l

,

bi > 0

for every i and consider the subspace generated by them. For a given natural number n let ~n be a scalar such that (a~/bn) > ctn > (an + 1)/(bn + 1). The vector a - ~nb attains its norm on the unit ball of m at the point {sign(at-ctnbi)}i=l and we have: sign(at - ~bt) = 1 sign(at - ~bi) = - 1

l= n + 1.

Choosing a scalar ~o such that ~o > (al/bl) the vector { a t - ~obt}t~l will attain its norm only at the point ( - 1, - 1, - 1, ...) of Sin. The smallest w*-closed sub-

1965]

ON FINITE DIMENSIONAL SUBSPACES OF BANACH SPACES

153

space of m which contains all the points of m found above is m itself, and this proves our statement. The space L l [ 0 , 1 ] fails also to have property (1). To see this it is enough to take the subspace generated by f l ( x ) = 1 and f2(x) = x and to use the fact that the characteristic functions of intervals form a total family over LI[0,1]. One may ask if the completion of a normed space satisfying one of the conditions (1), (2), (3) must satisfy this condition. The answer is negative since the linear subspace of 11 generated by the unit vectors is a member of d 3 while its completion fails to belong even to ~¢tTHEOREM 2. The spaces lp 1 < p < oo, where p is not an even integer, do not belong to the class ,~2. THEOREM 3. / f (S,E,p) is a measure space, the real space Lp(S,Y~,#)= Lp with p an even integer belongs to d 2 . In proving these theorems we shall constantly make use of the known facts that the function f ( t ) of Lp(p > 1) atains its norm on the unit ball of Lq(1/p+ 1/q = 1) at the point I f -P " f ( t ) p-lsign f(t) and only at this point. We shall refer everywhere to f ( t ) P - l s i g n f ( t ) since this function belongs to the same subspaees of Lq as f l 1-p. f ( t ) P-lsignf(t) does. Proof of Theorem 2. Let n be any integer greater than p. (If p is not an integer n may be any integer greater than 1). We shall consider the two dimensional subspace E n of l~, spanned by an = (1, 1,... 1) and bn = (1,2,3, ...,n) and we shall show that there is no proper subspace of lq which contains all the points of the unit ball having a support hyperplane generated by an element of En. Assume that n is an odd integer. Denote P,(2) = (I + ).i) p-I = I 1 + 2liP-1

1 < i < n.

Since a, + 2bn e E, for any real 2, in order to prove the assertion it will be enough to display n independent vectors of the the form {Pl(2) sign (1 + 2), P2(2) sign (1 + 22),...,P,(2)sign(1 + n2)). We shall give to 2 n different values submitted to the conditions:

2i>0

l 0;=

these inequalities prove (8) for n = 1. Again, from (6) follows that for large enough t and therefore

¢(20 < 2¢(t)

g i f t ) < 1 - ¢(2t) t 2t On the other hand, f~k+ 1)(X) is non decreasing by (5). Thus another application of the Mean Value Theorem will yield 1

ftk+x)[1- ~'~] 2t

t

=

2

"

t

> O, r

..

~

,

1965]

A NOTE ON ERLANG'S FORMULAS

159

hence afortiori

f(k) [1

t~) ]

f(k+~)[l_flp(t)] >

2~b(t)- ~b(2t) 2k

(2t)k- 1

t

> O.

tk

Thus we proved that if (8) holds for n = k - 1 then it holds also for n = k. 3. Proof of formulas (2). We prove the validity of the formulas (2) under assumption that the stream of calls began at the moment to = 0, i.e., no(0) = 1 (The proof would be only a little more difficult for arbitrary preliminary data). Let v(t) denote the probability that the service of a call that occurs in the time interval [0, t) will be determined in the same interval. It will be shown that

v(t) = 1 - -i-

(9) and therefore

H(x)dx

v(t) is independent of the number of calls which occur in the interval

[0, t). For the proof, let us assume that in the interval [0, t) exactly m calls occured and let r/be the moment that one of the calls, taken at random, appeared. Clearly

v(t) = PQI + ~ < t),

(10)

where ~ is the duration of service of a call. In [6] it was shown that if the stream is stationary and without after-effects then, whatever is m, the random variable r/is uniformly distributed in the interval [0, t). Since by our assumptions the variables t/and ~ are independent, equation (9) is easily deduced from (3) and (10). Denote by P,,(t) the probability that exactly m calls will occur in the interval [0, t). Then, by the above remarks, oo

n~(t) = ~,

Pm+k(t)("'~k)[1-v(t)]kv'(t)

m=O

or nk(t) _ [ 1 - v(t)] k £ vm(t). k[ m=O m!(m+k)!Pm+k(t)" Consider the generating function of the stream oo

F(t, x) = ~, Pm(t)x ra,

( I x ] ~ 1).

m=O

Since (m + k) lPm+k(t) =

8m+kF(t,O) 0xm+~

it is easy to see that the probabilities 7~k(t)

~m ( ~kF(t,x) ) -

~x m .

can also

dx k

[~=o,

be expressed by

160

D. MEJZLER

n (t)

[September

= [ 1 - v(t)] k t~kF[t,v(t)]

k!

Oxk

Finally, if we put (11)

q~(t) = f / H(x)dx,

we conclude from (9) that

qbk(t) 10kF[t,l--qS(t)/t] (12)

nk(t) =

k!

t~

~x k

It is known, [1], that the generating function of a stationary stream without after-effects has the form

r(t,x) = exp{2t[f(x) - 1]},

(13)

where 2 > 0 and p~ > 0, ( ~i~ z P~ = 1) are constants and oo

f(x) = ~, pix',

(14)

(Ix[ < 1).

i=l

Moreover, the intensity of the stream is given by GO

/t = ~ ipi = 2f'(1).

(15)

i=1

Assuming that the intensity is finite we get (16)

1 < f ' ( 1 ) < co.

It is easy to see that the partial derivatives of the generating function (13) are given by (17)

OkF(t'x) = F(t,x) {[2tf'(x)] k + k~- I (2t)m[ ~A(iD'",ir,) ~xk

m=l

~I s=l

fti')(x)]},

where the inner summation is over various systems of positive integers (il, "', i,) that satisfy (18)

il + ... +im = k,

and A(il, ..., ira) are constants that depend only on the indices (iz, "', ira). (Note that it ts possible to give the more detailed formula

k (2t)r~ ~ fti,)(X ) dkF(t.__._k_L,X)ox= k ! F(t, x),~=1 ~ . {Z ~=~1= is .' "' where the summation is over all the ordered systems of positive integers satisfying (18); for our purpose formula (17) is sutficient). Taking into account that in view of (18)

19651

A NOTE ON ERLANG'S FORMULAS

fl

16i

ti'- 1 = i k-ra,

s=l

we conclude by (12) and (17) that (19)

kt nk(t)=F(t,x) ~{[2f'(x)] k +'-'~___l~.rtt[~A(il,."" im)fi: : , f(~s'(X)II t'.-' JJ'

where we put for brevity x =

1

~(t) t

Our functions f ( x ) and ~b(t) satisfy all the conditions of the Lemma: f ( x ) by (14) and (16), and ~b(t) by (4) and (11). Since f(1) = 1, from (6), (7) and (13) follows that

limF[t,l--~-]

=e -~''°)

--- e - ' .

On the other hand, if 1 _< m _< k - 1, then every system of positive integers (il, "", ira) that satisfies equation (18) contains at least one integer i > 2. Therefore, because of (8), &

lira | 1

t-*oo s=l

tis-1

= 0

for all the systems (il, "--, ira) that paticipate in (19). Thus the second term in the braces of the expression (19), being a finite combination of terms that tend to zero, also tends to zero. This proves formulas (2), since lim ckk(t) = sk t--~ O0

and

REMARK. Formulas (2) can be applied to a slightly more general situation. Let a finite or countable number of stationary streams without after-effects xi(t) with intensity /~i (i = 1,2, ...) enter a service system consisting of n = ov lines. Assume that the distribution of the duration of the call service is the same for all calls in the same stream but this distribution depends on the index i and it may vary from one stream to another. Denote by st the mean duration of the service of a call which belongs to the stream x~(0, and assume that st < oo for every i, and ¢o

i=l

oo

i=I

I62

D. MEJZLER

Since the total stream x(t)= ~,~1 x~(t) is also finite, stationary and without after-effects, then formulas (2)hold also for it. It is clear that #and sare determined by co

1

oo

REFERENCES 1. A. Ya. Khintehine, Mathematical methods in the theory ofqueueing (translated from the Russian), Charles Griffin, London, 1960. 2. A. K. Erlang, Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges, Post Office Electrical Engineer's Journal 10 (1917-18), 189-197. 3. A. Ya. Klaimchin¢, Erlang's formulas in the theory of mass service, Theory of Probability and its Applications (an English translation of the Soviet Journal) "/(1963), No. 3, 320-325. 4. B. A. Sevast'yanov, An ergodic theorem for Markov processes and its applications to telephone systems with refusals, Theory of Probability and its Applications (an English translation of the Soviet Journal) 2 (1957), No. 1,104--112. 5. R. Fortet, Calcul des probabititds, Centre National de la Recherche Scientifique, Paris, 1950. 6. D. Mejzler, On a characteristic property of stationary streams without after-effect J. Appl. Prob. 2, (1965), 2 455--461. THE HEBREWUNIVERSITYOF JERUSALEM

SETS OF CONSTANT WIDTH IN FINITE DIMENSIONAL BANACH SPACES BY

H. G. EGGLESTON ABSTRACT

In Euclidean space a set of constant width has the property that it is not a proper subset of any set of the same diameter. The converse implication is also true. Here we show that if Euclidean is replaced by n-dimensional Banach space the direct statement is true, but the converse statement is false. Attention is drawn to the problem of characterising those Banach spaces of finite dimension for which the converse is true. Introduction. In Euclidean n-dimensional real space, a set X is said to be of constant width, if and only if, it is convex, compact, and the distance apart of any two parallel support hyperplanes is constant (i.e. the same whichever pair of parallel support hyperplanes we consider). A number of equivalent properties are known of which the most important (in Euclidean space) is that a set X is of constant width if and only if it is complete, that is to say, if Y is a set of which X is a proper subset then the diameter of Yexceeds that of X. Instead of"complete", which is used in many other ways, we shall use the phrase "diametrically maximal".

[4] The object of this note is to develop analogous properties in n-dimensional Banach spaces [see 2.3]. In particular we show, in contradiction to accepted belief [see 1], that the concepts "constant width" and "diametrical maximality" are not equivalent in every Banach space. It is always the case that constant width implies diametrical maximality. The reverse implication is not necessarily true even when the unit sphere of the space is both smooth and rotund. It is not easy to see what conditions on the space will ensure equivalence of these two concepts but a simple example can be given of a non Euclidean space in which this equivalence holds. Many of these equivalent conditions have been known previously (See [2]). Notation. Denote an n-dimensional Banach space by B" and its unit ball by S. For two parallel hyperplanes nl, n2 in B n the width between nx and n2 is twice the largest number 2 such that a set obtained from 2S by translation is contained in the strip bounded by nl and nz-A convex compact subset X of B n is of constant width if and only if the width between each pair of parallel support hyperplanes of X is constant. We shall use the phrase " o f constant width" to imply that X Received October 12, 1965. 163

164

H.G. EGGLESTON

(September

is convex and compact. The vector domain of a set X, denoted by Xv is the set of all vectors of the form xl - x2 where Xx and x2 vary indopendently in X. We use + between sets to denote vector sum. Thus Xv = X + ( - 1 ) X . Also the width of A + B between two parallel support hyperplanes n~, It2 is the sum of the widths of A and of B between corresponding pairs of hyperplanes all parallel to ltl and it2. Finally (A + B)v = Av + By. Except where the contrary is expressly stated every set is assumed to be convex and compact and to contain interior points. The origin of Bn, the centre of S, is denoted by 0. We use the same symbol for points and for vectors. The symbol xy will be used either for the line xy or the segment xy or the length of the segment xy. We shall use B(S) to indicate the Banach space with the central convex set S as its unit sphere.

1. Some properties equivalent to "constant width". (A) X (assumed to be convex and compact) is of constant width if and only if for some 2 > 0 X v = 2S. If X is of constant width then so are ( - 1 ) X and Xv = X + ( - 1 ) X . In any case X v is central thus it is sufficient to prove that a central set of constant width is a sphere. These two properties of Xv mean that for some fixed 2 > 0, 2S is supported by each pair of parallel support hyperplanes of Xv. Since a convex compact set is uniquely defined by its support hyperplanes, X v is 2S. On the other hand if X v is 2S then X v is of constant width. But the width of X between any pair of parallel support hyperplanes 7q, n2 is equal to the width of ( - 1 ) X between support hyperplanes parallel to 7tI and 7~2, and thus is half the width of Xv between two support hyperplanes parallel to ~ and n2. Hence X is of constant width. (B) A subset X of B" has the support intersection property if, and only if, given any pair of parallel support hyperplanes n~ and n2 of X, and one of the support hyperplanes of S, say ~r, parallel to nl and It2, then to any point p ~ n n S we can find x ~ n l r ~ X and x2en 2 n X s u c h that line x~x2 or x2x~ is parallel to the line op. X is of constant width if and only if it has the support intersection property. If X is of constant width then for some 2 > OX v = AS. Given nl,n2 and n all parallel hyperplanes of which the first two support X and the last S, let p ~ n n S. Then 2p e Xv and thus 2p = x~ - x2 where xx ~ X, x2 e X. Also x~ and x2 belong one each to n~ and ~2 (for 2p is a frontier point of Xv and there is a hyperplane of support to Xv at 2p parallel to n). Thus X has the support intersection property. Support next that X (assumed to be compact and convex) has the support intersection property. Let p be a point of the frontier of S, and let the half ray terminating at 0 and containing p meet the frontier of X v in q. If the hyperplane

1965]

CONSTANT WIDTH IN FINITE DIMENSIONAL BANACH SPACES

165

supports S at p then 3 xt, x2 on the frontier of X such that line xlx2 is parallel to op and there exist hyperplanes supporting X at xl, x2 and parallel to n. By definition X v contains the point xl - x2 and it is a frontier point of X r (hence it is q or - q ) through which passes a hyperplane parallel to n supporting X v. Thus S and X v are two central convex sets such that any half ray meets their frontiers at p and q respectively and if p lies on support hyperplane n of S then q lies on a parallel support hyperplane, of X v. We show that this implies that for some 2 > 0 Xv = 2S. It is convenient to consider B" as represented in Euclidean space E" with an appropriate metric. It is sufficient to show that if 0 Plq t and 0 P2q2 are two half rays terminating at 0 with PD P2 on the frontier of S and qtq2 on the frontier of Xv then 0 Pl [0 ql = 0 P 2 / O q2. The plane , = plane of o PlP2, o qlq2 meets S in a set say S' and X v in a set X~. The sets S', X v as two dimensional convex sets have the same property as do S and Xv. For if p is any point on the frontier of S' in z and h is a support line to S' at p in z, then there exists a support hyperplane nx to S at p meeting z in h. If o p meets frontier of Xv in q then there exists a support hyperplane of Xv at q parallel to nl and thus meeting z in a line parallel to h. This line supports X~, at q. In what follows for the remainder of this paragraph we consider only subsets of z. Select a fixed half line ox through o and measure angles in a fixed sense from ox. Let h(O) be the half line through o making an angle 0 with ox. 0 is the angle in E" which with the appropriate metric represents B ". Let h(O) meet the frontier of S' in p(O) and that of X~, in q(O). Denote the length (in E") op(O) by f(O) and oq(O) by g(O). Then if 1 0 1 - 02[ < n and K = sup f(0), ¢

(1)

If(00 - f(02)I < p(OOp(02)
0

(7)

x ~ (a, b).

The converse is also valid provided the differential operator in (7) is suitably interpreted (see Karlin and Studden (3) chap. 11 and Ziegler [6] for further details). Consider now the intersection cone f f a = 5 + n [(~,=o~(Uo, U~,...,u,)]

(8)

where cg+ denotes the cone of continuous non-negative functions defined on (a, b). It is proved in [3] that ~ belongs to ~a if and only if q~ is infinitely continuously differentiable, ~(x) > 0 on (a, b) and (7) holds for n = 0,1,.... Thus, the convex cone ¢ga coincides with the class of G.A.M. functions. We quote the following Taylor-type formula needed later. Let f(x) be any n + 1 times continuously differentiable function defined on (a, b) such that limx.~+ [d"f (x)] /dx "exists for each n(*). Then (9)

f(x) =

~b,(x; t)L,f(t) dt + ~

pk(a+)uk(x)

k=0

where

po(a +) - f(a +) Dk- ,... Dof(a +) Wo(a) pk(a +) = wk(a) , k = 1,2,-.and

[0 (10)

~Pn(X;t) :

J I

I

a<x 0 which may depend on dp. Then g(t) - 0 for t e [a, b). Proof.

By a result proved in [6'], rn

¢ ' ( t ; x ) e c~+ r3[k~--=o~(U0' "",Us)'],

for

n > m

and therefore

g(t) = lim

fb

n "-* oO ,d a

~n(t; X) dpn(x) e cg + n [

5 W(Uo,'", Uk)].

k = O

178

SAMUEL KARLIN AND ZVI ZIEGLER

[September

This holds independently of m, and therefore g(t) ~ ~a. As pointed out previously every member of ~a is automatically of class Coo(a, b) and moreover p,(a + ;g) exists for all n. Now suppose to the contrary that g(t) ~ 0 on [a, b); then there exists a maximal interval connected to a on which g(t) = 0. We denote this interval by [a, c*], c* < b and c* exceeds a by virtue of the hypothesis of the lemma. Since g(t) = 0 for t e [a, c*] and g(t) ~ C°°(a, b), the representation formula (9) applied to g(t) (with respect to the interval (c*, b)) reduces to

b

g(t) =

fc

~b,(t;x) dp,(x; g).

Invoking the assertion of the lemma for g(t), we infer that

(16)

g(t) = lim n-+oo

~cbq~.(t;x) dp.(x; g) = 0

t~[c*,c* +6)

*

for some 6 > 0. This conclusion is in contradiction to the definition of c* and the proof of the lemma is complete. We now prove that the hypothesis of the lemma is satisfied. Consulting (9), we see that f o r a _ < x _ < t < d = < b

c)(d) :>

=

fx d(~.(d;Odp.(~;~b) fx ddp.(d;Ow.+x(Op.+x(Od~

> p.+l(x)

fx dc~.(d;Ow.+l(Od~

>-_ p,,+l(x) ft d(a.(d;Ow.+x(Od~. Moreover, from the definition of qb.(d; 0

(17)

f/

(Pn(d;Own+x(Od~ >-

(

I-I mi(t;d)

i=o

"

(n+l)!

where

mi(t;d)=

min t 0. THEOREM 3. Let {W,}o satisfy the requirements of Theorem 1. A signed

measure d# belongs to the dual of ~A if and only if (22)

u~d# > 0

i = 0,1, ...

Proof. We know that u~ e ~°a, i = 0,1, ... so that (22) is certainly necessary. The validity of (13) easily implies that the inequality (22) is also sufficient. REFERENCES 1. D. V. Widder, TheLaplace transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, New Jersey, 1941. 2. I. I. Hirschman, Jr. and D. V. Widder, The convolution transform, Princeton University Press, Princeton, New Jersey, 1954. 3. S, Karlin and W. Studden, Tchebycheff Systems with Applications in analysis and Statistics, Interscience, New-York, to appear in 1966. 4. G. P61ya, On the mean value theorem corresponding to a given linear homogeneous differential equation, Tram. Amer. Math. Soc. 24 1922, 312-324. 5. S. Bochner, Harmonic analysis and the theory of probability, University of California Press, Berkeley and Los Angeles, 1955. 6. Z. Ziegler, Generalized convexity cones, to appear in Pac. J. Math. 7. S. Karlin and Z. Ziegler, Tchebycheffian spline functions, submitted to SIAM Journal. STANFORD UNIVERSITY, STANFORD, CALIFORNIA TECHNION--IsRAELINSTITUTEOF TECHNOLOGY, HAIFA~ ISRAEL