$L^\infty$ estimates in optimal mass transportation

We show that in any complete metric space the probability measures $\mu$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $\nu$ living on the support of $\mu$ is bounded below by a positive function of the $L^\infty$ transportation distance between $\mu$ and $\nu$. The function giving the lower bound depends only on the lower bound of the $\mu$-measures of balls centered at the support of $\mu$ and on the cost function used in the optimal transport. We obtain an essentially sharp form of this function. In the case of strictly convex cost functions we show that a similar estimate holds on the level of optimal transport plans if and only if the support of $\mu$ is compact and sufficiently close to being geodesic. We also study when convergence of compactly supported measures in $L^p$ transportation distance implies convergence in $L^\infty$ transportation distance. For measures with connected supports this property is characterized by uniform lower bounds on the measures of balls centered at the supports of the measures or, equivalently, by the Hausdorff-convergence of the supports.