Non-branching geodesics and optimal maps in strong CD(K,{\infty})-spaces
  We prove that in metric measure spaces where the entropy functional is K-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map.   The results are applicable in metric measure spaces having Riemannian Ricci-curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci-curvature bounded from below by some constant. 
Tapio Rajala and Karl-Theodor Sturm
math.DG math.MG