Applying twice a minimax theorem

Abstract

Here is one of the results obtained in this paper: Let $X, Y$ be two convex sets each in a real vector space, let $J:X\times Y\to {\bf R}$ be convex and without global minima in $X$ and concave in $Y$, and let $\Phi:X\to {\bf R}$ be strictly convex. Also, assume that, for some topology on $X$, $\Phi$ is lower semicontinuous and, for each $y\in Y$ and $\lambda>0$, $J(\cdot,y)$ is lower semicontinuous and $J(\cdot,y)+\lambda\Phi(\cdot)$ is inf-compact. Then, for each $r\in ]\inf_X\Phi,\sup_X\Phi[$ and for each closed set $S\subseteq X$ satisfying $$\Phi^{-1}(r)\subseteq S\subseteq \Phi^{-1}(]-\infty,r])\ ,$$ one has $$\sup_Y\inf_SJ=\inf_S\sup_YJ\ .$$