Convex Compact Sets that Admit a Lower Semicontinuous Strictly Convex Function

We study the class of compact convex subsets of a topological vector space which admit 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 many exposed points where a strictly convex lower semicontinuous function is continuous. As a consequence, every set in the above class is the closed convex hull of its exposed points.