Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendible geodesics

This paper provides rates of convergence for empirical barycenters of a Borel probability measure on a metric space under general conditions. Our results are given in the form of sharp oracle inequalities. Our main assumption connects ideas from metric geometry to the theory of empirical processes and is studied in two meaningful scenarios. The first one is a geometrical constraint on the underlying space referred to as $(k,\alpha)$-convexity, compatible with a positive upper curvature bound in the sense of Alexandrov. The second scenario considers the case of a nonnegatively curved space on which geodesics, emanating from a barycenter, can be extended. While not restricted to this setting, our results are discussed in the context of Wasserstein spaces.