Fundamental polytopes of metric trees via parallel connections of matroids
arXiv:1612.05534 · doi:10.1016/j.ejc.2020.103098
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.
20 pages, 2 Figures, 1 Table. Exposition improved, references and new results (last section) added