We study the Wasserstein distance between two measures $\mu,\nu$ which are mutually singular. In particular, we are interested in minimization problems of the form $$W(\mu,\A)=\inf\big\{W(\mu,\nu)\ :\ \nu\in{\mathcal A}\big\}$$ 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 $$\min\big\{P(B)+kW(A,B)\ :\ |A\cap B|=0,\ |A|=|B|=1\big\},$$ where $k>0$ is a fixed constant, $P(A)$ is the perimeter of $A$, and both sets $A,B$ may vary.

### On the Wasserstein distance between mutually singular measures

#### Abstract

We study the Wasserstein distance between two measures $\mu,\nu$ which are mutually singular. In particular, we are interested in minimization problems of the form $$W(\mu,\A)=\inf\big\{W(\mu,\nu)\ :\ \nu\in{\mathcal A}\big\}$$ 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 $$\min\big\{P(B)+kW(A,B)\ :\ |A\cap B|=0,\ |A|=|B|=1\big\},$$ where $k>0$ is a fixed constant, $P(A)$ is the perimeter of $A$, and both sets $A,B$ may vary.
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Buttazzo, Giuseppe; Carlier, Guillaume; Laborde, Maxime
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/939838
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