Mesh ratios for best-packing and limits of minimal energy configurations

TL;DR

This paper investigates the geometric properties of best-packing configurations on compact metric spaces, establishing bounds on mesh-separation ratios and characterizing unique minimal energy configurations, especially on the sphere.

Contribution

It provides new estimates for mesh-separation ratios in best-packing configurations and characterizes the unique limit configuration for five points on the sphere.

Findings

The mesh-separation ratio is bounded above by 1 for certain configurations.

A specific 5-point configuration on the sphere is identified as the unique limit of minimal energy configurations.

The square-base pyramid configuration achieves the bound of the mesh-separation ratio.

Abstract

For $N$-point best-packing configurations $\omega_N$ on a compact metric space $(A,\rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\omega_N,A)$, which is the quotient of the covering radius of $\omega_N$ relative to $A$ and the minimum pairwise distance between points in $\omega_N$. For best-packing configurations $\omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s\to \infty$, we prove that $\gamma(\omega_N,A)\le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $\rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $\omega_5^*$, that is the limit (as $s\to \infty$) of 5-point $s$-energy minimizing configurations. Moreover, $\gamma(\omega_5^*,S^2)=1$.