A Differential Characterisation of the Minimax Inequality

We prove the following result: let $K\subseteq \mathbb{R}^N$ be convex with nonempty interior, $X$ a topological space and $f\colon K\times X\to\mathbb{R}$ be concave and u.s.c. in the first variable and coercive and l.s.c. in the second. Then the (perturbed) strict minimax inequality $$\sup_{\lambda\in K}\inf_{x\in X}f(\lambda,x)+g(\lambda)<\inf_{x\in X} \sup_{\lambda\in K}f(\lambda,x)+g(\lambda),$$ for some continuous concave $g\colon K\to\mathbb{R}$, is equivalent to the following condition on superdifferentials: if $F(\lambda)=\inf_X f(\lambda, x)$, for some $\lambda\in\mathring{K}$ $$\partial F(\lambda)\setminus \bigcup_{\substack{x\in X\\ f(\lambda, x) =F(\lambda)}}\partial f(\lambda, x)\neq\emptyset.$$ As an application of this differential characterisation we prove a generalised version of a theorem of Ricceri, a criterion of regularity for marginal functions, and the fact that to check whether some perturbed minimax inequality holds, one can test with affine perturbation only.