Sala Seminari (Dip. Matematica).<\/p>\n
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 $$W(\\mu,{\\cal A})=\\inf\\big\\{W(\\mu,\\ u)\\ :\\ \\ u\\in{\\cal A}\\big\\}$$ where $\\mu$ is a given probability and ${\\cal A}$ is contained in the class $\\mu^\\perp$ of probabilities that are singular with respect to $\\mu$. Several cases for ${\\cal A}$ are considered; in particular, when ${\\cal A}$ 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 $$\\min\\big\\{P(B)+kW(A,B)\\ :\\A\\cap B=0,\\A=B=1\\big\\},$$ where $k>0$ is a fixed constant, $P(A)$ is the perimeter of $A$, and both sets $A,B$ may vary.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 $$W(\\mu,{\\cal A})=\\inf\\big\\{W(\\mu,\\ u)\\ :\\ \\ u\\in{\\cal A}\\big\\}$$ where $\\mu$ is a…<\/p>\n