L∞ estimates in optimal mass transportation

We show that in any complete metric space the probability measures μ with compact and connected support are the ones having the property that the optimal transportation distance to any other probability measure ν living on the support of μ is bounded below by a positive function of the L∞ transportation distance between μ and ν. The function giving the lower bound depends only on the lower bound of the μ-measures of balls centered at the support of μ 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 onl…