Topological Properties of the Space of Convex Minimal Usco Maps

Abstract

We investigate the space of convex minimal usco maps from a Tychonoff space to the space of real numbers. Its elements are set-valued maps that are important e.g. in the study of subdifferentials of convex functions. We show that if the underlying space is normal, convex minimal usco maps can be approximated in the Vietoris topology by continuous functions. Using the strong Choquet game we prove complete metrizability of the space of convex minimal usco maps equipped with the upper Vietoris topology. We also study first countability, second countability and other properties of the (upper) Vietoris topology on this space.