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Fundamental polytopes of metric trees via parallel connections of matroids
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.Comment: 20 pages, 2 Figures, 1 Table. Exposition improved, references and new
results (last section) adde
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