Minimax Equalities by Reconstruction of Polytopes

Given a quasi-concave-convex function f : X × Y → R defined on the product of two convex sets we would like to know if infY supX f = supX infY f . In [4] we showed that that question is very closely linked 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 ∈ 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 1.1, gives necessary and sufficient conditions for a subset-stable family F of subsets of a convex set X to verify X ∈ F. Theorem 1.1 leads to some nontrivial minimax equalities, some of which are presented here: Theorems 1.3, 1.5, 3.4, 3.5, 4.1 and their corollaries. Further applications of our method to minimax equalities will be carried out in a forthcoming paper [5].