  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.
