On the Wasserstein distance between mutually singular measures

We study the Wasserstein distance between two measures mu, v which are mutually singular. In particular, we are interested in minimization problems of the form W(mu, A) = inf {W(mu, v) : v is an element of A}, where mu is a given probability and A is contained in the class mu(perpendicular to) of probabilities that are singular with respect to mu. Several cases for A are considered; in particular, when.A consists of L-1 densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem min {P(B) + kW(A, B) : vertical bar A boolean AND vertical bar = 0, vertical bar A vertical bar = vertical bar B vertical bar = 1}, where k > 0 is a fixed constant, P(A) is the perimeter of A, and both sets A, B may vary.