Abstract In the present work we prove that minimizers of the Wasserstein- ℋ 1 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.
João Miguel Machado (Thu,) studied this question.