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Journal of Convex Analysis 19 (2012), No. 1, 185--199
Copyright Heldermann Verlag 2012

A Differential Characterisation of the Minimax Inequality

Sunra J. N. Mosconi
Dip. di Matematica e Informatica, UniversitÓ di Catania, Viale A. Doria 6, 95125 Catania, Italy


\def\R{\mathbb{R}} We prove the following result: let $K\subseteq \R^N$ be convex with nonempty interior, $X$ a topological space and $f\colon K\times X\to\R$ 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 $g\colon K\to\R$, 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.

Keywords: Minimax inequality, concave functions, marginal functions, multiple solutions to variational problems, nonlinear eigenvalues.

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