Minimax equalities by reconstruction of polytopes

Given a quasi-concave-convex function f : X x Y --> (R) over bar defined on the product of two convex sets we would like to know if inf(Y) sup(X) f = sup(X) inf(Y) f. In [4] we showed that that question is very closely linked to the following "reconstruction" problem: given a polytope (i.e. the conv ex hull of a finite set of points) X and a family IF of subpolytopes of X, we would like to know if X f IF, knowing that any polytope which is obtaine d by cutting an element of F with a hyperplane or by pasting two elements o f IF along a common facet is also in F. Here, we consider a similar "recons truction" problem for arbitrary convex sets. Our main geometric result, The orem 1.1, gives necessary and sufficient conditions for a subset-stable fam ily F of subsets of a convex set X to verify X is an element of F. Theorem 1.1 leads to some nontrivial minimax equalities, some of which are presente d here: Theorems 1.3, 1.5, 3,4, 3.5, 4.1 and their corollaries. Further app lications of our method to minimax equalities will be carried out in a fort hcoming paper [5].