  We investigate barycenters of probability measures on Gromov hyperbolic
spaces, toward development of convex optimization in this class of metric
spaces. We establish a contraction property (the Wasserstein distance between
probability measures provides an upper bound of the distance between their
barycenters), a deterministic approximation of barycenters of uniform
distributions on finite points, and a kind of law of large numbers. These
generalize the corresponding results on CAT(0)-spaces, up to additional terms
depending on the hyperbolicity constant.
