Existence and consistency of Wasserstein barycenters

Based on the Fréchet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean . We prove the existence of Wasserstein barycenters of random probabilities defined on a geodesic space ( E , d ). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the probability measures or of a growing set of probability measures.