Uniqueness of maximum three-distance sets in the three-dimensional Euclidean space

2013/09/09 by Masashi Shinohara, Shinohara, Masashi
Mathematics · #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:52C35

paper · pdf · doi:10.48550/arxiv.1309.2047

10 pages, 4 figures

arxiv created 2013/09/09 · arxiv updated 2013/09/10

Abstract

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distances between two distinct points in X. Einhorn and Schoenberg conjectured that the vertices of the regular icosahedron is the only 12-point three-distance set in ℝ³ up to isomorphism. In this paper, we prove the uniqueness of 12-point three-distance sets in ℝ³.

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