Compact convex sets that admit a lower semicontinuous strictly convex function

TL;DR

This paper characterizes compact convex sets that admit a lower semicontinuous strictly convex function, showing they can be embedded in a strictly convex dual Banach space and identifying points of continuity.

Contribution

It establishes a connection between such convex sets and embeddings into strictly convex dual Banach spaces, and analyzes the continuity points of the functions.

Findings

Compact convex sets with the property can be embedded in strictly convex dual Banach spaces.

Existence of exposed points where the function is continuous.

Provides structural insights into the geometry of these convex sets.

Abstract

We study the class of compact convex subsets of a topological vector space which admits a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with its weak$^*$ topology. In addition, we find exposed points where a strictly convex lower semicontinuous function is continuous.