On a minimax theorem: an improvement, a new proof and an overview of its applications

Abstract

Theorem 1 of [14], a minimax result for functions $f:X\times Y\to \mathbf{R}$, where $Y$ is a real interval, was partially extended to the case where $Y$ is a convex set in a Hausdorff topological vector space ([15], Theorem 3.2). In doing that, a key tool was a partial extension of the same result to the case where $Y$ is a convex set in ${\mathbf{R}}^n$ ([7], Theorem 4.2). In the present paper, we first obtain a full extension of the result in [14] by means of a new proof fully based on the use of the result itself via an inductive argument. Then, we present an overview of the various and numerous applications of these results.

Draws from arXiv:1312.5715, arXiv:math/0703931, arXiv:1406.1026, arXiv:1201.1574, arXiv:1409.5919, arXiv:1510.05036

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