What question did this study set out to answer?

To present a variant of the minimax theorem that accommodates non-convex functions and demonstrates its application to remoteness.

June 17, 2026

Partially non-convex minimax theorem and applications to unique remoteness of sets and functions

Key Points

To present a variant of the minimax theorem that accommodates non-convex functions and demonstrates its application to remoteness.
Established a variant of the minimax inequality considering functions that may take infinite values.
Identified subsets where the functions are convex, specifically detailing the behavior across these subsets.
Demonstrated that the minimax inequality holds under these new conditions.
Provided a framework for assessing unique remoteness in mathematical sets and functions.

Abstract

We establish the following variant of the minimax inequality maxy∈Binfx∈Af(y,x)≥infx∈Asupy∈B0f(y,x), where, unlike the classical minimax theorem, the functions f(y,⋅) may not be convex for all y∈B but only for some of them (those in a possibly smaller set B0), while the function f is allowed to take infinite values. An application to the unique remoteness of sets and functions is given.

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