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

In the present work we prove that minimizers of the Wasserstein- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">ℋ</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {H¹} 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.