Minimax Equalities by Reconstruction of Polytopes

Given a quasi-concave-convex real-valued function $f: X\times Y \to \overline{\mathbb{R}}$ defined on the product of two convex sets we would like to know if $\inf_Y \sup_X f = \sup_X \inf_Y f$. 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 $\mathbb{F}$ of subpolytopes of X, we would like to know if $X \in \mathbb{F}$, knowing that any polytope which is obtained by cutting an element of $\mathbb{F}$ with a hyperplane or by pasting two elements of $\mathbb{F}$ along a common facet is also in $\mathbb{F}$. 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 $\mathbb{F}$ of subsets of a convex set X to verify that $X \in \mathbb{F}$. 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]