Local Maxima of the Potential Energy on Spheres

Let S (d) be a unit sphere in a"e (d+1), and let alpha be a positive real number. For pairwise different points x (1),x (2), . . . ,x (N) a S (d) , we consider a functional E (alpha) (x (1),x (2), . . . ,x (N) ) = I pound (i not equal j) ||x (i) - x (j) ||(-alpha) . The following theorem is proved: for alpha a parts per thousand yen d - 2, the functional E (alpha) (x (1),x (2), . . . ,x (N) ) does not have local maxima.