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

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 ${\mathbb{R}}^3$ up to isomorphism. In this paper, we prove the uniqueness of 12-point three-distance sets in ${\mathbb{R}}^3$.