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

Let SdSd denote the unit sphere in the Euclidean space Rd+1(d≥1). Let N   be a natural number (N≥2)(N≥2), and let ωN≔{x1,…,xN}ωN≔{x1,…,xN} be a collection of N   distinct points on SdSd on which the minimal Riesz s  -energy is attained. In this paper, we show that the points x1,…,xNx1,…,xN are well-separated for the cases d-1≤s