A Proof of a Dodecahedron Conjecture for Distance Sets

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 ℝ^3 is 20, and every 5-distance set in ℝ^3 with 20 points is similar to the vertex set of a regular dodecahedron.