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Владикавказский математический журнал Январь-март, 2004, Том 6, Выпуск 1
UDC 517.98
ON MINIMAX THEOREMS FOR SETS CLOSED IN MEASURE
A. V. Bukhvalov and A. Martellotti
The paper is dedicated
to the memory of Yuri Abramovich,
who was dear friend for both of us.
This article is devoted to the Ky Fan minimax theorem for convex sets closed in measure in L1. In
general, these sets do not carry any formal compactness properties for any reasonable topology.
Famous Fan's minimax theorem, stated in its generality, claims that for a function y) defined on 1x7 (where X and Y have no linear or convex structures) which satisfies a mild convex-concave-like conditions, the minimax equality holds provided the set X is compact (see [5] for a simple proof which we will use later for our generalization).
Bukhvalov and Lozanovsky invented in [7] the Optimization Without Compactness (OWC) technique for sets in L1 (see [6] for a survey of the current state of art; an elementary exposition is given in [11]). Roughly speaking norm bounded convex sets in L1, which are closed with respect to convergence in measure, have many properties usually associated with compact sets only.
Our main goal is to derive a minimax theorem without compactness (Section 1) and further to extend it with milder convexity assumptions (Section 2). Section 3 briefly describes possible applications.
1. Main Theorem
Our assumptions will be the following (all sets below are expected to be non empty). Let (T, S, /i) be a cr-finite measure space, and let X be a convex, norm-bounded set in Ll([i). Let Y be an arbitrary set and #;IxF-iRbea function such that
(a) for each y ei the function x H- $(x,y) is convex;
(b) for each t € [0,1] and for every yi,y2 € Y there exists ys £Y with
y3) > J/i) + (1 — y2)
for every x € X.
Condition (b) is more general than concavity. The latter is too restrictive for many-applications. It is said that $(x,y) is concave-like in y. This class has a much wider area of applications (see below).
© 2004 Bukhvalov A. V., Martellotti A.
Theorem 1. Suppose that X, Y and # satisfy the conditions above. Suppose that X is closed in measure and is lower semicontinuous in measure on X for each y G F. Then
min sup #(2;, y) = supmin#(a;, y). (1)
xtXyeY yeYxtx
We will start by justifying 'min' instead of 'inf' in (1). The following well-known consequence of [7] is stated in [13], [11].
Lemma 1. Let X be a convex norm-bounded set in Ll([i) which is closed in measure. Let f: X -ï M be a quasi-convex function which is lower semicontinuous in measure. Then f attains its minimum in X.
Lemma 2. Let X, Y and # be as in Theorem 1. Then the function
f(x) := sup{#(œ,y): y G Y}
is quasi-convex and lower semicontinuous in measure on X. Proof of Theorem 1. Let
p := minsup#(a;,y).
x£X yeY
If p = —co there is nothing to prove since the inequality
inf sup $(x,y) ^ sup inf $(x,y) xexyeY yeYxex
always holds.
Let a be any real such that a < p. By our assumptions each set
C(y):={xeX: $(x,y)i^a}
is norm bounded, convex and closed in measure for any y. Since the assumption a < p implies that the whole intersection of such sets is empty, from Theorem 1.3 in [6] we deduce that there exist yi,... ,yn G Y such that C(y\) D C(y2) D ... D C(yn) = 0. If we assume that
a ^ min sup $(x,yi),
lsCisCra
then there should exist xq G X such that a > $(xo,yi) for all yi, i = 1, ...,n. Hence we would obtain that xq G C(yi), i = 1,... ,n. This contradiction shows that
a < min sup $(x,yi).
IsCisCw
Consider now the set
E := {(z,r) £Rn+1 : (3a; G X) $(x,yi) ^ r + zu ¿ = 1,...,«}. The set E is convex. Indeed, if
®(x,yi) ^ r + Zi, i = l,...,n, ®(x,yi) ^f + Zi, i = 1,... ,n;
then, for t € [0,1] and for x = tx + (1 — t)x we have
$(®,yi) = + (1 -t)%yi) < i$(®,y0 + (1 -i)$(I,yi)
< i(r + Zi) + (1 -i)(F + fi) = ir+ (1 + + (1 -i)fj.
=r =z.
Then, the point
(0,1 + max $(®,yj) J € K"+1
is interior to E for any x € X. Indeed, if |r| < e and \zi\ < e, i = 1,,,,, n, then, putting a; = m) we get #(3?, m) ^ 1 + #(3?, m) + Zi + r provided |zi + r\ < 2e ^ 1. Also,
by construction (0, a) £ E. Indeed, #(3?, yi) ^ a, i = 1,,,,, n, for some x € X implies that
min sup $(x,yi) si a
which contradicts the choice of a.
Now we can apply the Separation Theorem to the point (0, a) 0 E and to the convex set E in the finite dimensional space M"+1: note that it is not difficult to prove that the set E is also closed, though we do not need this to apply the Separation Theorem. So we can find a vector (Ai,..., Ar) ^ 0 with
n
^A iZi + rr^tra ((z,r)eE). (2)
¿=1
It is clear that E + M"+1 C E. If it were r < 0 then, taking r > 0 one would come to a contradiction with (2). This proves that r > 0. Analogously, if Aj < 0 for some i, then taking the corresponding entry Zi > 0 we would come to a contradiction with (2). Again this proves that Aj > 0 for each i = 1, ...,n. Note that actually r > 0 as the point (0,1 + maxi^„ $(x,yi)) lies in the interior of E. Indeed, if r = 0, then ^ 0
for all (z,r) € E. Taking (z,r) with zi negative, and sufficiently close to 0, we find
n
^izi< 0
1=1
since Ai > 0, although (z, r) is still a point in E.
The point ($(x,yi) + r, —r) lies in E for all x € X and all r € M. Indeed, $(x,yi) ^ $(x,yi) + r — r. Then, the inequality (2) implies that
n
¿(#(®,yi)+r)+r(^r) >ra. (3)
¿=1
Since r > 0, dividing (3) by r we have
n ( n \
E^1 ]r><*> (4)
1=1 \i=l /
for all x € X and all r£l.
Since rGlis arbitrary, we derive that ^ = 1. So, (4) can be rewritten as
¿=1
n A-
¿=1
where on the left-hand side we have a convex combination of the n points $(x,yi). Then condition (b) implies, by induction, that there exists y € Y such that $(x,y) > a for all x € X, whence
supmin#(a;,y) ^ a.
y£Y x^X
Since a is arbitrary underneath p, we achieve
min sup y) = supmin#(a;, y). > xtXyeY yeYxtx
In Section 3 we will discuss some possible application of Theorem 1; for instance, it is usual to connect minimax equality with results concerning the consistency of a system of inequalities. We explicitly present a result in this direction, which can be derived from (the dual version of) Theorem 1.
Proposition 1. Let H be a convex set of concave functionals defined on a set X c Ll(p) which is closed, norm-bounded and closed in measure. Also assume that each f £ H is upper semicontinuous in measure on X. Suppose that for every / eW there exists ijGl such that f(xf) > 0. Then there exists a point xq € X such that f(xo) ^ 0 for all /eW.
Remarks. (1) Since OWC technique is true for more general situation then L1, e. g., for perfect Banach Function Spaces and their vector-valued generalizations when the corresponding Banach space of values is reflexive (see [6]), then the results here and below extend to that more general setting.
(2) Using OWC technique, Levin derived a minimax theorem (see [6; Theorem 4.1]) but he confined himself with convex-concave setting, which is both easier for the proof and not much interesting in applications. Actually, only the approach from [5] gave us the tools for the current general result.
2. Weaker Convexity Conditions
In the literature connected with minimax relationships and related topics, many-generalizations of the notion of convexity have been proposed.
Definition 1. A function # : X x Y —M is finite convex-like with respect to x if, for each finite set {yi,..., yn} C Y, each pair xi,x2 el and every t € [0,1], an element ^el exists, such that
#(®3,yi) <i$(®i,yi) + (l-i)$(®2,yi) (5)
for every i = 1,... ,n.
This definition has been introduced by Granas and Liu [9] in the following more general way:
Definition 2. A function # : X x Y —M is midpoint finite convex-like with respect to x if, for each finite set {yi,... ,y„} C Y and each pair X\,X2 € X, an element x^ € X exists, such that
1
$(®3,yi) < g + (6)
for every i = 1,... ,n.
It is quite clear that if condition (a) of Section 1 is replaced by (ai) # is finitely convex-like with respect to x; (a2) for every y ei the map x H- $(x,y) is quasi-convex, then Theorem 1 is still true.
It is less trivial to note that the result remains true if (ai) is replaced by the midpoint finite convex-likeness with respect to a;. In fact the following result holds:
Proposition 2. Let X be a convex, norm-bounded set in Ll([i), Y any non-empty set arid #: X x Y M be such that
(i) for each y 6 7, the map x H- $(x,y) is lower semicontinuous in measure;
(ii) for each y G Y, the map x H- $(x,y) is quasi-convex; (iii) # is finitely midpoint convex-like with respect to x.
Then # is finitely convex-like with respect to x. <1 Let {yi,..., yn} C Y and x\, G X be fixed.
Claim 1: For every q G Q(2) (the set of dyadic rational numbers in [0,1]) there exists xq G X such that
for each i = 1,..., n.
This is a standard iterative argument. The proof is that of ([17; Lemma 3.2]). Claim 2: Let (Cn)
n be a non increasing sequence of convex subsets of L1 (p) which are closed in measure. Then
P| Cn = {x G L1 (p): x = (p)- lim®„, xn G Cn} =: B.
n
The inclusion f)nCn C. B is immediate. Conversely, since the sequence is decreasing, and each set Cn is closed in measure, it is clear that B C Cn for every n.
Claim 3: # is finitely convex-like with respect to x.
Let t G [0,1] be fixed, and let (tk)k be a sequence in Q(2) with tk t. From Claim 1 we can determine Xk G X such that
$(®fc,yi) < tkHxuyi) + (1 - ifc)$(®2,yi)
for every i = 1,..., n and every k G N.
Since (xk) C X, and X is by assumption norm bounded, by Theorem 1.4 in [6] there are: a sequence of integers 1 = k\ < < ..., a sequence of non-negative numbers (Xp)p with hi-i-i — l hi-i-i — l
Aj = 1 and such that the sequence gn = ^ AjXj converges ¡jl-a. e. to an element
j=ki j=ki x G E.
For each i = 1,... ,n and every p, define the number
M(i,p) = max{tj$(xuyi) + (1 - tj)®(x2,yi), j = kp,.. .,kp+1 - 1} and note that, since tj t, then
]im[tj^(xi,yi) + (l-tj)^(x2,yi)] = t^(xi,yi) + (l-t)^(x2,yi) =■■ Ti
for every i = 1,..., n. Hence limp M(i,p) = Tj, for i = 1,..., n.
Set now
Cij, = C(M(i,p)) = {x€X: $(s,y) i = l,...,n, p G N}
and observe that (i) and (ii) ensure that each C¿)P is convex and closed in measure. Hence, since for each integer k in [kp,kp+1 — 1] the corresponding Xk G CiiP for i = l,...,n then gp G Ci)P for i = 1,..., n and p G N.
By a diagonal argument, we can find a subsequence {pr,r G N} such that (M(i,p)) is monotone for i = 1,..., n. Let
J\ = {i G {1,... ,n} such that (M(i,pr))r is non decreasing}, J2 = {i G {1,... ,n} such that (M(i,pr))r is non increasing}.
Now, for i G Ji the sequence (CijPr)r is non decreasing with respect to r, and CitPr C C(r¿). Hence G C(r¿). Since C(r¿) is closed in measure, and (gPr /it-converges to x, x £ C(r¿), namely,
Ti = t(f)(xuyi) + (1 - í)#(®2,y¿)
for i G J\.
For i G J2, the sequence (CitPr)r is non increasing; as gPr G CiiPr, by Claim 2, x G f]r ^.Pr-But, as it is easily seen, in this case
f| Ci;PrCC(n). >
r
Remark. Note that Proposition 2 remains true in any topological vector space, if X has a sort of the Mazur property, that is, if from every sequence in X we can construct a sequence of convex combinations of terms with far numbers that converges in the topology of E.
3. Applications
Here we present some directions of possible applications of minimax theorems given above and OWC technique in general. In some cases we have developed results, in the other just an understanding of some links.
1. Duality for tp(X,Y) spaces. For Banach Function Spaces X and Y on the same measure space and a concave function ip(x,y) of two variables one can define the space (p(X,Y) (see the details in [14]). The principal duality equality established by Lozanovskii reads as tp(X,Y)' = ip(X',Y'), where prime stands for Banach Function Space dual and (p stands for a suitable convolution operator. This result has been refined both in isomorphic and isometric form by Lozanovskii himself (see [15, 16]), Berezhoi [1, 2] and Reisner [19]. Note that Lozanovskii used OWC technique (separation theorem). Reisner was able to reduce the problem for power function </? to finite-dimensional functions; to do this he had to use Ky Fan's minimax theorem with convex-concave-like assumptions as in our Theorem 1. For Theorem 1 we do not need the machinery of such reduction dealing directly with infinite-dimensional setting. Here we are able to prove all types of duality theorems quoted above.
2. Markov-Kakutani theorem. In [3] a version of the Markov-Kakutani theorem was presented for commuting families of affine contractions on norm bounded convex sets in L1, which are closed in measure. In [10] a similar result was published for weakly continuous affine mappings (not necessarily contractions). Both proofs are based on OWC technique. Using OWC technique and [20] we are able to prove an analogous result for families of mappings,
which are continuous in measure. We do not know the correspondence between the last two results.
3. Stochastic games. Using Theorem 1 and OWC technique it is possible to establish very general infinite-dimensional results on existence of equilibrium in stochastic games (cf. [4; Appendix]).
4. Point estimates in statistics. It is possible to establish some existence theorems for minimax point estimates in mathematical statistics (cf. [12]).
5. Miscellanea. In factorization Theorem 109 [18] we can drop the condition of reflexivity for the Orlich space Lg because of Proposition 1.
In [8], the proof for existence of an equivalent martingale measure is given. The major step there is to move from an easily constructed element of the second dual to an element from L1. Though we do not have complete proof by means of OWC we would like to note that this issue is obviously related to our topic (see [8; Appendix 1]).
Acknowledgements. Main theorem and its applications were presented to the Stieltjes Mathematical Society (the Netherlands) in September 1999 by the first author. He is grateful the organizers, and in particular to Professor Ben de Pagter, for their kind invitation.
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Received by the editors April 5, 2004-
Bukhvalov Alexander Vasil'evich, <J>.-m. h. Russia, St. Petersburg, School of Management St. Petersburg State University E-mail: [email protected]
Martellotti Anna
Department of Mathematics and Informatics
University of Perugia
Via Vanvitelli 1, Perugia 06123, Italy
