On the infimum of the upper envelope of certain families of functions

TL;DR

This paper develops a general scheme using a recent minimax theorem to precisely determine the infimum of a class of functions defined by supremums over linear combinations of continuous functions, with applications to explicit minimizers.

Contribution

It introduces a novel framework for calculating the exact infimum of complex functions involving supremums, extending previous minimax results to broader classes of functions.

Findings

Exact value of the infimum for a large class of functions derived.

Existence of explicit functions achieving the infimum under compactness conditions.

Framework applicable to various problems involving supremums of linear combinations.

Abstract

In this paper, given a topological space $X$, an interval $I\subseteq {\bf R}$ and five continuous functions $\varphi, \psi, \omega :X\to {\bf R}$, $\alpha, \beta:I\to {\bf R}$, we are interested in the infimum of the function $\Phi:X\to ]-\infty,+\infty]$ defined by $$\Phi(x)=\sup_{\lambda\in I}(\alpha(\lambda)\varphi(x)+\beta(\lambda)\psi(x))+\omega(x)\ .$$ Using a recent minimax theorem ([5]), we build a general scheme which provides the exact value of $\inf_X\Phi$ for a large class of functions $\Phi$. When additional compactness conditions are satisfied, our scheme provides also the existence of (explicitly detected) functions $\gamma, \eta:X\to {\bf R}$ such that, for some $\tilde x\in X$, one has $$\gamma(\tilde x)\varphi(\tilde x)+\eta(\tilde x)\psi(\tilde x)+\omega(\tilde x)=\inf_{x\in X}(\gamma(\tilde x)\varphi(x)+\eta(\tilde x)\psi(x)+\omega(x))\ .$$