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

TL;DR

This paper improves a minimax theorem for functions with convex set domains, provides a new proof using induction, and reviews its diverse applications across mathematical fields.

Contribution

It offers a full extension of a minimax theorem with a novel inductive proof and surveys its broad range of applications.

Findings

Extended the minimax theorem to convex sets in topological vector spaces.

Developed a new proof technique based on induction.

Highlighted numerous applications of the extended theorem.

Abstract

Theorem 1 of [14], a minimax result for functions $f:X\times Y\to {\bf R}$, 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 ${\bf R}^n$ ([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.