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

For N-point best-packing configurations ωN on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ωN,A), which is the quotient of the covering radius of ωN relative to A and the minimum pairwise distance between points in ωN. For best-packing configurations ωN that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ωN,A)≦1 and this bound can be attained even for the sphere. In the particular case when N=5 on S2 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 $\omega_{5}^{*}$, that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, $\gamma(\omega_{5}^{*},S^{2})=1$.