Journal of Convex Analysis 07 (2000), No. 2, 335--352
Copyright Heldermann Verlag 2000
Minimax Equalities by Reconstruction of Polytopes
Gabriele H. Greco
Dip. di Matematica, Università di Trento, 38050 Povo, Italy
Charles D. Horvath
Dép. de Mathématiques, Université de Perpignan, 66860 Perpignan, France
Given a quasi-concave-convex real-valued function f: X×Y --> R defined on the product of two convex sets we would like to know if inffY supX f = supX inffY f. We showed in another paper [A reconstruction of polytopes by convex pastings, to appear in Mathematika] that this question is very closely related to the following "reconstruction" problem: given a polytope (i.e. the convex hull of a finite set of points) X and a family F of subpolytopes of X, we would like to know if X is an element of F, knowing that any polytope which is obtained by cutting an element of F with a hyperplane or by pasting two elements of F along a common facet is also in F. Here, we consider a similar reconstruction problem for arbitrary convex sets.
Our main geometric result, Theorem A, gives necessary and sufficient conditions for a subset-stable family F of subsets of a convex set X to verify that X is an element of F. Theorem A leads to some nontrivial minimax equalities, some of which are presented here: Theorems 1, 2, 7, 8, 9 and their corollaries. Further applications of our method to minimax equalities will be carried out in a forthcoming paper of the authors [Toward a geometric theory of minimax equalities, to appear in Optimization].
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