Least Wasserstein distance between disjoint shapes with perimeter regularization

TL;DR

This paper proves the existence of optimal pairs of disjoint shapes minimizing a combined perimeter and Wasserstein distance, addressing a fundamental question in geometric measure theory across all dimensions.

Contribution

It establishes the existence of global minimizers for a shape optimization problem involving perimeter and Wasserstein distance in any dimension, for all p and positive lambda.

Findings

Existence of minimizers proven for all dimensions and parameters.

Addresses a previously open question by Buttazzo, Carlier, and Laborde.

Results extend the theory of shape optimization with Wasserstein metrics.

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.