The Triangular Bi-Pyramid Minimizes a Range of Power Law Potentials

Abstract

Combining a brilliant obserbation of A. Tumanov with our computational approach to Thomson's 5-electron problem, we prove that the triangular bi-pyramid is the unique global minimizer for the Rieze potential R_s(r) = sign(s) r^{-s} amongst all configurations of 5 points on the unit sphere, provided that s in (-2,0) or s in (0,13]. The lower bound is sharp and the upper bound is pretty close to the presumed sharp cutoff of about 15.040908. I hope to reach the sharp cutoff in a sequel paper. The discussion section of this paper has a brief sketch of the approach I will take in the sequel to deal with exponents in the range [13,15.04.0809...].