Minimax Equalities by Reconstruction of Polytopes

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 inff_(Y) sup_(X) f = sup_(X) inff_(Y) 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]