# Non-branching geodesics and optimal maps in strong $CD(K,\infty)$-spaces

**accepted**
**year:** 2012

**journal:** Calc. Var. Partial Differential Equations

**abstract:** 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.

The paper is available on the
cvgmt preprint server.