Characterization of finite metric space by their isometric sequences

2018/02/01 by Hirasaka, Mitsugu, Shinohara, Masashi
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1802.06097

Abstract

Let (X,d) be a finite metric space with |X|=n. For a positive integer k we define A_k(X) to be the quotient set of all k-subsets of X by isometry, and we denote |A_k(X)| by a_k. The sequence (a₁,a₂,…,a_n) is called the isometric sequence of (X,d). In this article we aim to characterize finite metric spaces by their isometric sequences under one of the following assumptions: (i) a_k=1 for some k with 2≤ k≤ n-2; (ii) a_k=2 for some k with 4≤ k≤ (1+√(1+4n))/(2); (iii) a₃=2; (iv) a₂=a₃=3. Furthermore, we give some criterion on how to embed such finite metric spaces to Euclidean spaces. We give some maximum cardinalities of subsets in the d-dimensional Euclidean space with small a₃, which are analogue problems on a sets with few distinct triangles discussed by Epstein, Lott, Miller and Palsson.

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