On the ARG MIN multifunction for lower semicontinuous functions

Abstract:

The epi-topology on the lower semicontinuous functions $L\left ( X \right )$ on a Hausdorff space $X$ is the restriction of the Fell topology on the closed subsets of $X \times R$ to $L\left ( X \right )$, identifying lower semicontinuous functions with their epigraphs. For each $f \in L\left ( X \right )$, let arg min $f$ be the set of minimizers of $f$. With respect to the epi-topology, the graph of arg min is a closed subset of $L\left ( X \right ) \times X$ if and only if $X$ is locally compact. Moreover, if $X$ is locally compact, then the epi-topology is the weakest topology on $L\left ( X \right )$ for which the arg min multifunction has closed graph, and the operators $f \to f \vee g$ and $f \to f \wedge g$ are continuous for each continuous real function $g$ on $X$.