Séminaire Lotharingien de Combinatoire, 93B.5 (2025), 12 pp.
Emanuele Delucchi, Lukas Kühne and Leonie Mühlherr
Combinatorial Invariants of Finite Metric Spaces and the Wasserstein Arrangement
Abstract.
In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called "Wasserstein polytopes" or "Kantorovich-Rubinstein polytopes" in the literature.
Answering a question posed by Vershik, we describe the stratification of the metric cone induced by the combinatorial type of these polytopes through a hyperplane arrangement. Moreover, we study its relationships with the stratification by combinatorial type of the injective hull (i.e., the tight span) and, in particular, with certain types of metrics arising in phylogenetic analysis. We also compute enumerative invariants in the case of metrics on up to six points.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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