Extremal arrangements of points and unit charges on a sphere: equilibrium configurations revisited

We review the formulation and solutions of a number of extremal problems associated with points and unit charges on the surface of a sphere in E 3 . For one of these problems, namely[Formula: see text]where d pq is the Euclidean distance between points P and Q and m is the number of points, we discuss the results for m ≤ 16 and 1 ≤ n ≤ ∞. For the cases m = 5, 11, 13–16 we find hitherto undiscovered solutions. Our solutions for m = 5 and 11 correct earlier results in the literature. We also sharpen the existing literature results for m = 7 and 10.