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

Abstract

In this paper, given a topological space $X$, an interval $I\subseteq \mathbf{R}$ and five continuous functions $\phi ,\psi ,\omega :X\to \mathbf{R}$, $\alpha ,\beta :I\to \mathbf{R}$, we are interested in the infimum of the function $\Phi :X\to ]-\infty ,+\infty ]$ defined by $$\Phi (x)=\underset{\lambda \in I}{\sup }(\alpha (\lambda )\phi (x)+\beta (\lambda )\psi (x))+\omega (x).$$ 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 $\gamma ,\eta :X\to \mathbf{R}$ such that, for some $\tilde{x}\in X$, one has $$\gamma (\tilde{x})\phi (\tilde{x})+\eta (\tilde{x})\psi (\tilde{x})+\omega (\tilde{x})=\underset{x\in X}{\inf }(\gamma (\tilde{x})\phi (x)+\eta (\tilde{x})\psi (x)+\omega (x)).$$