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.
Lipschitz polytopes of finite metric spaces are weighted digraph polyhedra in the sense of Joswig and Loho @cite_20 , who give some general results about dimension, face structure and projections [ 2.1, 2.2, 2.6] JL but mostly focus on the case of braid cones'' which does not apply to our context. We close by mentioning that the polyhedra considered, e.g., in the above-mentioned work of Hirai [Formula (4.1)] Hirai are different from the Lipschitz polytopes we consider here: in fact, such polyhedra are (translated) zonotopes for all split-decomposable metrics [Remark 4.10] Hirai , while -- for instance -- the Lipschitz polytope of any split-decomposable metric on @math points is only a zonotope if the associated split system is compatible.
@cite_20
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This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of Rd. This leads to natural cell decompositions of the tropical projective space TPmind−1. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007).