Optimal (Formula presented)-Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

Abstract

This paper inquires into the concavity of the map (Formula presented) from the integers (Formula presented) into the minimal average standardized Riesz pair-energies (Formula presented) of (Formula presented)-point configurations on the sphere (Formula presented) for various (Formula presented). The standardized Riesz pair-energy of a pair of points on (Formula presented) a chordal distance (Formula presented) apart is (Formula presented), (Formula presented), which becomes (Formula presented) in the limit (Formula presented). Averaging it over the (Formula presented) distinct pairs in a configuration and minimizing over all possible (Formula presented)-point configurations defines (Formula presented). It is known that (Formula presented) is strictly increasing for each (Formula presented), and for (Formula presented) also bounded above, thus “overall concave.” It is (easily) proved that (Formula presented) is even locally strictly concave, and that so is the map (Formula presented) for (Formula presented). By analyzing computer-experimental data of putatively minimal average Riesz pair-energies (Formula presented) for (Formula presented) and (Formula presented), it is found that the map (Formula presented) (Formula presented) is locally strictly concave, while (Formula presented) is not always locally strictly concave for (Formula presented): concavity defects occur whenever (Formula presented) (an (Formula presented)-specific empirical set of integers). It is found that the empirical map (Formula presented), is set-theoretically increasing; moreover, the percentage of odd numbers in (Formula presented) is found to increase with (Formula presented). The integers in (Formula presented) are few and far between, forming a curious sequence of numbers, reminiscent of the “magic numbers” in nuclear physics. It is conjectured that these new “magic numbers” are associated with optimally symmetric optimal-log-energy (Formula presented). A list of interesting open problems is extracted from the empirical findings, and some rigorous first steps toward their solutions are presented. It is emphasized how concavity can assist in the solution to Smale’s (Formula presented)th Problem, which asks for an efficient algorithm to find near-optimal (Formula presented)-point configurations on (Formula presented) and higher-dimensional spheres.