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Mesh ratios for best-packing and limits of minimal energy configurations. (English) Zbl 1349.31004

Summary: 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 \(\gamma(\omega_N, A)\leqq 1\) 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, \(\gamma(\omega_5^\ast, S^2) = 1\).

MSC:

31C20 Discrete potential theory
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
57N16 Geometric structures on manifolds of high or arbitrary dimension
52A40 Inequalities and extremum problems involving convexity in convex geometry
28A78 Hausdorff and packing measures
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