On the ARG MIN multifunction for lower semicontinuous functions

Abstract

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