On barycenters of probability measures

TL;DR

This paper characterizes barycenters of Radon probability measures on convex subsets, especially focusing on the space of finite signed measures and probability measures, providing simple criteria for locally compact spaces.

Contribution

It offers a new characterization of barycenters of Radon measures on convex sets, including the space of finite signed measures and probability measures, with simple conditions for locally compact spaces.

Findings

Characterization of barycenters of Radon measures on convex sets.

Special case analysis for measures on metric compact spaces.

Simple criteria for barycenters in locally compact spaces.

Abstract

A characterization is presented of barycenters of the Radon probability measures supported on a closed convex subset of a given space. A case of particular interest is studied, where the underlying space is itself the space of finite signed Radon measures on a metric compact and where the corresponding support is the convex set of probability measures. For locally compact spaces, a simple characterization is obtained in terms of the relative interior.