An order inequality characterizing invariant barycenters on symmetric cones

This paper is concerned with invariant and contractive barycenters on the Wasserstein space of probability measures on metric spaces of non-positive curvature, where the center of gravity, also called the Cartan barycenter, is the canonical barycenter on Hadamard spaces. We establish an order inequality of probability measures on partially ordered symmetric spaces of non-compact type, namely symmetric cones (self-dual homogeneous cones), characterizing the Cartan barycenter among other invariant and contractive barycenters. The derived inequality and partially ordered structures on the probability measure space lead also to significant results on (norm) inequalities including the Ando-Hiai inequality for probability measures on symmetric cones.