Optimal Riemannian distances preventing mass transfer

We consider an optimization problem related to mass transportation: given two probabilities f þ and f on an open subset WHRN, we let vary the cost of the transport among all distances associated with conformally flat Riemannian metrics on W which satisfy an integral constraint (precisely, an upper bound on the L1-norm of the Riemannian coe‰cient). Then, we search for an optimal distance which prevents as much as possible the transfer of f þ into f : higher values of the Riemannian coe‰cient make the connection more di‰cult, 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.