Inserted: 29 jul 2012
Last Updated: 15 jul 2013
Journal: Calc. Var. Partial Differential Equations
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.
Keywords: Ricci curvature, metric measure spaces, branching geodesics