Minimal Discrete Energy on the Sphere

We investigate the energy of arrangements of $N$ points on the surface of a sphere in ${\bold R}^3$, interacting through a power law potential $V=r^\alpha$, $-2 <\alpha <2$, where $r$ is Euclidean distance. For $\alpha=0$, we take $V=\log(1/r)$. An area-regular partitioning scheme of the sphere is devised for the purpose of obtaining bounds for the extremal (equilibrium) energy for such points. For $\alpha=0$, finer estimates are obtained for the dominant terms in the minimal energy by considering stereographical projections on the plane and analyzing certain logarithmic potentials. A general conjecture on the asymptotic form (as $N \to \infty$) of the extremal energy, along with its supporting numerical evidence, is presented. Also we introduce explicit sets of points, called “generalized spiral points", that yield good estimates for the extremal energy. At least for $N \leq 12,\,000$ these points provide a reasonable solution to a problem of M. Shub and S. Smale arising in complexity theory.

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Published 1 January 1994