Absence of loops for the Wasserstein-$\mathcal{H}^1$ problem: the concentration/blow-up argument

TL;DR

This paper proves that minimizers of a specific Wasserstein problem are trees under certain conditions, advancing understanding of optimal transport structures.

Contribution

It establishes that minimizers are trees for the Wasserstein-$\

Findings

Minimizers are trees when the target measure is finitely supported.

Minimizers are trees when the target measure has a bounded density.

Abstract

In the present work we prove that minimizers of the Wasserstein-$\mathscr{H}^1$ problem, introduced recently by Chambolle et. al., are trees in two cases: when the target measure is a sum of finitely many Dirac masses or when it has a bounded density.