DOI: 10.1515/acv-2025-0071 ISSN: 1864-8258

Absence of loops for the Wasserstein-ℋ 1 problem: The concentration/blow-up argument

João Miguel Machado
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Abstract

In the present work we prove that minimizers of the Wasserstein-

ℋ 1 {\mathscr{H}^{1}}
problem, introduced recently in [A. Chambolle, V. Duval and J. a. M. Machado, One-dimensional approximation of measures in Wasserstein distance, J. Éc. polytech. Math. 12 2025, 101–145], are trees in three cases: When the target measure is a sum of finitely many Dirac masses, when it has a bounded density, when it is a mixture of an atomic measure and a measure with bounded density, and when it is a mixture of both these cases.

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