Calculus of Variations and Geometric Measure Theory

M. Novack - I. Topaloglu - R. Venkatraman

Least Wasserstein distance between disjoint shapes with perimeter regularization

created by topaloglu1 on 10 Aug 2021
modified on 12 Oct 2022


Published Paper

Inserted: 10 aug 2021
Last Updated: 12 oct 2022

Journal: J. Funct. Anal.
Volume: 284
Number: 1
Pages: 109732
Year: 2023

ArXiv: 2108.04390 PDF


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