On a distance representation of Kantorovich potentials

We address the question of how to represent Kantorovich potentials in the mass transportation (or Monge-Kantorovich) problem as a signed distance function from a closed set. We discuss geometric conditions on the supports of the measure f+ and f− in the Monge-Kantorovich problem which ensure such a representation. Finally, we obtain, as a by-product, the continuous differentiability of the potential on the transport set.