# Least Wasserstein distance between disjoint shapes with perimeter regularization

created by topaloglu1 on 10 Aug 2021

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Submitted Paper

Inserted: 10 aug 2021
Last Updated: 10 aug 2021

Year: 2021

Abstract:

We prove the existence of global minimizers to the double minimization problem $\inf\Big\{ P(E) + \lambda W_p(\mathcal{L}^n\lfloor \, E,\mathcal{L}^n\lfloor\, F) \colon E \cap F = 0, \, E = F = 1\Big\},$ where $P(E)$ denotes the perimeter of the set $E$, $W_p$ is the $p$-Wasserstein distance between Borel probability measures, and $\lambda > 0$ is arbitrary. The result holds in all space dimensions, for all $p \in [1,\infty),$ and for all positive $\lambda$. This answers a question of Buttazzo, Carlier, and Laborde.