Slopes of Kantorovich potentials and existence of optimal transport maps   in metric measure spaces
  We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps. 
Luigi Ambrosio and Tapio Rajala
math.MG math.DG