Explore barycenters of probability measures on Gromov hyperbolic spaces in this 51-minute lecture. Delve into the development of convex optimization in metric spaces, examining the contraction property in terms of the Wasserstein distance and a form of law of large numbers for stochastic approximation of barycenters. Discover how these findings generalize corresponding results on CAT(0)-spaces, with additional terms dependent on the hyperbolicity constant.