Least Wasserstein distance between disjoint shapes with perimeter regularization

Abstract

We prove the existence of global minimizers to the double minimization problem \[ \inf\Big\{ P(E) + λW_p(\mathcal{L}^n \lfloor \, E,\mathcal{L}^n \lfloor\, F) \colon |E \cap F| = 0, \, |E| = |F| = 1\Big\}, \] 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.

This is a post-peer-review, pre-copyedit version of an article published in Journal of Functional Analysis. The final authenticated version is available online at: https://doi.org/10.1016/j.jfa.2022.109732