On the Wasserstein distance between mutually singular measures

We study the Wasserstein distance between two measures $mu, u$ which are mutually singular. In particular, we are interested in minimization problems of the form $$W(mu,A)=infig{W(mu, u) : uin{mathcal A}ig}$$ where $mu$ is a given probability and ${mathcal A}$ is contained in the class $mu^perp$ of probabilities that are singular with respect to $mu$. Several cases for ${mathcal A}$ are considered; in particular, when ${mathcal A}$ 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 $$minig{P(B)+kW(A,B) : |Acap B|=0, |A|=|B|=1ig},$$ where $k>0$ is a fixed constant, $P(A)$ is the perimeter of $A$, and both sets $A,B$ may vary.