DOI: 10.68381/jca07017 ISSN: 0944-6532
Minimax Equalities by Reconstruction of Polytopes
Gabriele H. Greco, Charles D. Horvath
Given a quasi-concave-convex real-valued function
f: X\times Y \to \overline{\mathbb{R}}
f
:
X
×
Y
→
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
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}
F
of subpolytopes of X, we would like to know if
X \in \mathbb{F}
X
∈
F
, knowing that any polytope which is obtained by cutting an element of
\mathbb{F}
F
with a hyperplane or by pasting two elements of
\mathbb{F}
F
along a common facet is also in
\mathbb{F}
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}
F
of subsets of a convex set X to verify that
X \in \mathbb{F}
X
∈
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]