Combinatorial invariants of finite metric spaces and the Wasserstein arrangement

TL;DR

This work develops a combinatorial and polyhedral framework to classify finite metric spaces by the combinatorial type of Kantorovich–Rubinstein (KRW) polytopes, a perspective tied to optimal transport and the injective hull (tight span). It introduces the Wasserstein arrangement, a hyperplane arrangement that refines the KRW-type fan and yields a finer stratification of the metric cone; the authors connect this to regular triangulations via the GKZ secondary fan and to phylogenetic metric substructures such as tree-like and circular-decomposable metrics. They establish incomparability between KRW-type stratifications and tight-span stratifications for general $n$, while identifying noteworthy subfans and establishing exact f-vector counts in the strict case. Computationally, they enumerate KRW polytope types for generic metrics on up to six points, using OSCAR, CountingChambers.jl, and TOPCOM, revealing rich combinatorial structures even at small sizes and highlighting the impact of symmetry on the counts. The paper thus provides a robust invariant framework for finite metric spaces with applications to phylogenetics and geometric statistics, and lays groundwork for further study of metric-space subdivisions and their enumerative properties.

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.

Figures (8)

• Figure 1: A hyperplane $H_{\mathbf{a,b}}$ with $k=3$ interpreted as cycle.
• Figure 2: Illustration for \ref{['ex:4pt']}. (a), (c): KRW polytopes of the metrics $\rho_1$, $\rho_2$. (b): Combinatorial type of the tight span. The horizontal edges are of length $1$ and $2$ for the metrics $\rho_1$ and $\rho_2$, respectively. All other edges are of length $1$.
• Figure 3: Schlegel diagram of the polar dual of the KRW polytope (i.e., the Lipschitz polytope) of the metric $\rho_1$ from \ref{['example_decomp']}.
• Figure 4: Sketch of the tight span of the metric $\rho_1$ in \ref{['example_decomp']}. The complex $E(\rho_1)$ has dimension $2$ and consists of four rectangular faces and five dangling edges. It has $14$ total vertices, five of which (the solid dots in the picture) are the image of the canonical embedding of the metric space into its injective hull. We have labeled some edges with a number corresponding to the distance of the edge's endpoints with respect to the distance in the injective hull, which is induced on $E(\rho_1)\subseteq \mathbb R^{5}$ by the uniform norm ("supremum-norm"). Edges that appear parallel in the picture have the same "length". See also \ref{['gendegen']}.
• Figure 5: Sketch of the tight span of the metric $\rho_2$ in \ref{['example_decomp']}. The complex $E(\rho_2)$ has $5$ two-dimensional cells (two pentagons, two quadrilaterals, one triangle) and five dangling edges. This metric is "generic of type II" in the sense of Bandelt and Dress BandeltDress.

Theorems & Definitions (106)

• Definition 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Example 2.4: Split pseudometrics
• Theorem 2.5: GordonPetrov, Theorem 3
• Definition 2.6
• Lemma 2.7
• proof
• Corollary 2.8