  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.
