  This work establishes fast rates of convergence for empirical barycenters
over a large class of geodesic spaces with curvature bounds in the sense of
Alexandrov. More specifically, we show that parametric rates of convergence are
achievable under natural conditions that characterize the bi-extendibility of
geodesics emanating from a barycenter. These results largely advance the
state-of-the-art on the subject both in terms of rates of convergence and the
variety of spaces covered. In particular, our results apply to
infinite-dimensional spaces such as the 2-Wasserstein space, where
bi-extendibility of geodesics translates into regularity of Kantorovich
potentials.
