On separation of minimal Riesz energy points on spheres in Euclidean spaces

Let S d denote the unit sphere in the Euclidean space R d + 1 ( d ≥ 1 ) . Let N be a natural number ( N ≥ 2 ) , and let ω N ≔ { x 1 , … , x N } be a collection of N distinct points on S d on which the minimal Riesz s-energy is attained. In this paper, we show that the points x 1 , … , x N are well-separated for the cases d - 1 ≤ s < d .