Local Maxima of the Potential Energy on Spheres

Abstract

Let S ^d be a unit sphere in ℝ ^d+1, and let α be a positive real number. For pairwise different points x ₁,x ₂, . . . ,x _N ∈ S ^d , we consider a functional E _α (x ₁,x ₂, . . . ,x _N ) = Σ _i≠j ||x _i − x _j ||^−α . The following theorem is proved: for α ≥ d − 2, the functional E _α (x ₁,x ₂, . . . ,x _N ) does not have local maxima.