An Extremal Property Of The Icosahedron

The purpose of this article is to prove that the product of all possible pairwise distances between twelve points located on the unit sphere in a 3-dimensional Euclidean space is not greater than 2 132 =5 30 , and this maximal value is attained on the vertices of a regular icosahedron inscribed in the sphere. 2 3 = 1g be the unit sphere in the 3-dimensional Euclidean space. For any x; y 2 R 3 we denote by xy the inner product of the corresponding vectors with origin at the zero and having x and y as endpoints; jxj := p xx.