Combinatorial invariants of finite metric spaces and the Wasserstein arrangement

TL;DR

This paper explores the combinatorial structure of Wasserstein polytopes associated with finite metric spaces, revealing their stratification, relationships with injective hulls, and enumerative properties for small point sets.

Contribution

It describes the stratification of the metric cone by polytope type, relates it to the injective hull stratification, and computes invariants for small metrics, advancing understanding of metric space invariants.

Findings

Stratification of metric cone by polytope type

Relationship between Wasserstein polytopes and injective hulls

Enumerative invariants for metrics on up to six points

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.