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 $\gamma ({\omega }_N,A)$, 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 ${\omega }_5^{\ast }$, that is the limit (as $s\to \infty$) of 5-point $s$-energy minimizing configurations. Moreover, $\gamma ({\omega }_5^{\ast },S^2)=1$.