Fundamental polytopes of metric trees via parallel connections of matroids

TL;DR

This paper explores the combinatorial structure of fundamental polytopes associated with metric trees, providing explicit formulas and characterizations for their face numbers and simplicial properties.

Contribution

It introduces a novel approach linking metric trees to matroid theory via parallel connections, offering explicit formulas and classifications.

Findings

Explicit formulas for face numbers of fundamental and Lipschitz polytopes

Characterization of metric trees with simplicial fundamental polytopes

Connection between metric trees and matroid theory

Abstract

We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010. In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid. We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we characterize the metric trees for which the fundamental polytope is simplicial.