Local Maxima of the Potential Energy on Spheres - Radchenko, D. V. (Ukrains’kyi Matematychnyi Zhurnal, 2013)
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.