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Least Wasserstein distance between disjoint shapes with perimeter regularization

Affiliation:	1. Department of Mathematics, The University of Texas at Austin, Austin, TX, United States of America;2. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, United States of America;3. Courant Institute of Mathematical Sciences, New York University, New York, NY, United States of America

Abstract:	We prove the existence of global minimizers to the double minimization problemP(E) denotes the perimeter of the set E, $W_{p}$ is the p-Wasserstein distance between Borel probability measures, and $λ > 0$ is arbitrary. The result holds in all space dimensions, for all $p \in [1, \infty)$ , and for all positive λ. This answers a question of Buttazzo, Carlier, and Laborde.

Keywords:	Nonlocal isoperimetric problem Global existence Wasserstein distance Perimeter regularization
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