Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations

Abstract

For $N$-point best-packing configurations $\omega_N$ on a compact metric space $(A, \rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\rho_N , A)$, 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 $\gamma(\omega_N , A) ≤ 1$ and this bound can be attained even for the sphere. In the particular case when $N = 5$ on $S^1$ 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, $\gamma(\omega^*_5, S^2) = 1$.