Some reverse inequality in optimal mass transportation

Controlling the ${\mathcal{W}}_{\mathrm{\infty }}$ Wasserstein distance by the ${\mathcal{W}}_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year later [14] framed it in the same inequality for more general costs which increase with the distance. In this paper, we prove this type of inequality for optimal transport problems with pointwise cost which is a decreasing function of the distance. We show, in particular, that there is a general framework that encompasses all the cases above.