Trace Convexity and Choquet Theory

Abstract

We study the notion of trace-convexity for functions and respectively, for subsets of a compact topological space. This notion generalizes both classical convexity of vector spaces, as well as Choquet convexity for compact metric spaces and provides an alternative description for the convexification for sets and functions. We show that the class of upper semicontinuous convex-trace functions attaining their maximum at exactly one Choquet-boundary point is residual and we obtain several enhanced versions of the maximum principle, including a multi-maximum principle for families of convex-trace functions, which generalize both the classical Bauer’s theorem as well as its abstract version in the Choquet theory. We illustrate our notions and results with concrete examples of three different types.