Some new minimax theorems for generalized convexity

TL;DR

This paper extends classical minimax theorems to a non-compact setting using two-functions and establishes equivalences with known inequalities, broadening the theoretical framework of generalized convexity.

Contribution

It introduces new two-functions minimax inequalities and shows their equivalence to Simons' inequality under general conditions, in a non-compact context.

Findings

Extended minimax inequalities for generalized convexity.

Proved equivalence between one-function minimax equality and Simons' inequality.

Applicable in non-compact settings.

Abstract

The aim of this article is to establish new two-functions minimax inequalities extending classical results such as Simons' minimax theorem. Our results will be proved in a non-compact setting. We also prove, under general conditions, that the one-function minimax equality is in fact equivalent to the well known Simons inequality. Some applications will be given.