2021/10/05 by Mathey-Prevot, Maxime, Valette, Alain
#05C05 #05C12 #46B85 #68R12 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2110.02115
We study the Wasserstein (or earthmover) metric on the space P(X) of probability measures on a metric space X. We show that, if a finite metric space X embeds stochastically with distortion D in a family of finite metric trees, then P(X) embeds bi-Lipschitz into ℓ1 with distortion D. Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans-Matsen \citeEvMat. We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).