Kantorovich type topologies on spaces of measures and convergence of barycenters

We study two topologies $ \tau_{KR} and \tau_K on the space of measures a completely regular generated by Kantorovich–Rubinshtein Kantorovich seminorms analogous to their classical norms in case metric space. The topology coincides with weak nonnegative bounded uniformly tight sets measures. A sufficient condition is given for compactness topology. show that logarithmically concave measures, some s $-convex stable convergence implies also obtain an efficiently verified barycenters Radon from sequence or net weakly converging locally convex As application it shown convergent holds without additional conditions. same true polynomial densities fixed degree respect