Optimal $$N$$ N -Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

Abstract

This paper inquires into the concavity of the map $$N\mapsto v_s(N)$$ from the integers $$N\ge 2$$ into the minimal average standardized Riesz pair-energies $$v_s(N)$$ of $$N$$ -point configurations on the sphere $$\mathbb {S}^2$$ for various $$s\in \mathbb {R}$$ . The standardized Riesz pair-energy of a pair of points on $$\mathbb {S}^2$$ a chordal distance $$r$$ apart is $$V_s(r)= s^{-1}\left( r^{-s}-1 \right) $$ , $$s \ne 0$$ , which becomes $$V_0(r) = \ln \frac{1}{r}$$ in the limit $$s\rightarrow 0$$ . Averaging it over the $$\left( \begin{array}{c} N\\ 2\end{array}\right) $$ distinct pairs in a configuration and minimizing over all possible $$N$$ -point configurations defines $$v_s(N)$$ . It is known that $$N\mapsto v_s(N)$$ is strictly increasing for each $$s\in \mathbb {R}$$ , and for $$s<2$$ also bounded above, thus “overall concave.” It is (easily) proved that $$N\mapsto v_{-2}^{}(N)$$ is even locally strictly concave, and that so is the map $$2n\mapsto v_s(2n)$$ for $$s<-2$$ . By analyzing computer-experimental data of putatively minimal average Riesz pair-energies $$v_s^x(N)$$ for $$s\in \{-1,0,1,2,3\}$$ and $$N\in \{2,\ldots ,200\}$$ , it is found that the map $$N\mapsto {v}_{-1}^x(N)$$ is locally strictly concave, while $$N\mapsto {v}_s^x(N)$$ is not always locally strictly concave for $$s\in \{0,1,2,3\}$$ : concavity defects occur whenever $$N\in {\mathcal {C}}^{x}_+(s)$$ (an $$s$$ -specific empirical set of integers). It is found that the empirical map $$s\mapsto {\mathcal {C}}^{x}_+(s),\ s\in \{-2,-1,0,1,2,3\}$$ , is set-theoretically increasing; moreover, the percentage of odd numbers in $${\mathcal {C}}^{x}_+(s),\ s\in \{0,1,2,3\}$$ is found to increase with $$s$$ . The integers in $${\mathcal {C}}^{x}_+(0)$$ 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 $$N$$ -point configurations on $$\mathbb {S}^2$$ . 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 $$7$$ th Problem, which asks for an efficient algorithm to find near-optimal $$N$$ -point configurations on $$\mathbb {S}^2$$ and higher-dimensional spheres.