Maximizers of nonlocal interactions of Wasserstein type

TL;DR

This paper proves that balls uniquely maximize a Wasserstein-based functional involving set interactions, using symmetrization and transport plan uniqueness, with a refined one-dimensional result.

Contribution

It establishes the uniqueness of ball maximizers for a Wasserstein interaction functional and provides a sharp quantitative refinement in one dimension.

Findings

Balls are the only maximizers of the functional.

Symmetrization-by-reflection technique is effective.

A sharp quantitative refinement is achieved in one dimension.

Abstract

We characterize the maximizers of a functional that involves the minimization of the Wasserstein distance between sets of equal volume. We prove that balls are the only maximizers by combining a symmetrization-by-reflection technique with the uniqueness of optimal transport plans. Further, in one dimension, we provide a sharp quantitative refinement of this maximality result.