%0 Journal Article
%T Non-branching geodesics and optimal maps in strong CD(K,{\infty})-spaces
%A Tapio Rajala
%A Karl-Theodor Sturm
%J Mathematics
%D 2012
%I arXiv
%X 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.
%U http://arxiv.org/abs/1207.6754v2