On the Wasserstein distance between mutually singular measures.

Summary We study the Wasserstein distance between two measures µ, ν which are mutually singular. In particular, we are interested in minimization problems of the form W (µ, A) = inf W (µ, ν) : ν ∈ A where µ is a given probability and A is contained in the class µ ⊥ of probabilities that are singular with respect to µ. 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) : |A ∩ B| = 0, |A| = |B| = 1 , where k > 0 is a fixed constant, P (A) is the perimeter of A, and both sets A, B may vary.