A proof of a Dodecahedron conjecture for distance sets

Abstract

A finite subset of a Euclidean space is called an $s$-distance set if there exist exactly $s$ values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in ${\mathbb{R}}^3$ is 20, and every $5$-distance set in ${\mathbb{R}}^3$ with $20$ points is similar to the vertex set of a regular dodecahedron.

16 pages, 7 figures