Fundamental polytopes of metric trees via parallel connections of matroids

• DOI: 10.1016/j.ejc.2020.103098
• Publication Date: 2020-03-25T16:38:47Z
• Authors: Emanuele Delucchi; Linard Hoessly
• Keywords: FOS: Mathematics; 05-XX (Primary), 92Bxx (Secondary), 54E35; Mathematics - Combinatorics; 0102 computer and information sciences; Combinatorics (math.CO); 0101 mathematics; 01 natural sciences

20 pages, 2 Figures, 1 Table. Exposition improved, references and new results (last section) added<br/>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.<br/>