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

For $N$-point best-packing configurations $ω_N$ on a compact metric space $(A,ρ)$, we obtain estimates for the mesh-separation ratio $γ(ω_N,A)$, which is the quotient of the covering radius of $ω_N$ relative to $A$ and the minimum pairwise distance between points in $ω_N$. For best-packing configurations $ω_N$ that arise as limits of minimal Riesz $s$-energy configurations as $s\to \infty$, we prove that $γ(ω_N,A)\le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $ρ$ 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 $ω_5^*$, that is the limit (as $s\to \infty$) of 5-point $s$-energy minimizing configurations. Moreover, $γ(ω_5^*,S^2)=1$.

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