Optimal Riemannian distances preventing mass transfer

We consider an optimization problem related to mass transportation: given two probabilities f(+) and f(-) on an open subset Omega subset of R-N, we let vary the cost of the transport among all distances associated with conformally. at Riemannian metrics on Omega which satisfy an integral constraint ( precisely, an upper bound on the L-1-norm of the Riemannian coefficient). Then, we search for an optimal distance which prevents as much as possible the transfer of f(+) into f(-): higher values of the Riemannian coefficient make the connection more difficult, but the problem is non-trivial due to the presence of the integral constraint. In particular, the existence of a solution is a priori guaranteed only on the relaxed class of costs, which are associated with possibly non-Riemannian Finsler metrics. Our main result shows that a solution does exist in the initial class of Riemannian distances.