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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 $sto infty$, 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 $omega_5^*$, that is the limit (as $sto infty$) of 5-point $s$-energy minimizing configurations. Moreover, $gamma(omega_5^*,S^2)=1$.
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In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+infty],$ the $N$-point $f$-best-packing constant $min{f(|x-y|), :, x,yin R^d}$, where the minimum is taken over point sets of cardinality $N.$ We also show that $$ N^{1/d}Delta_d^{-1/d}-2le D_d(N)le N^{1/d}Delta_d^{-1/d}, quad Nge 2,$$ where $Delta_d$ is the maximal sphere packing density in $R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $Ntoinfty$.
We investigate the asymptotic behavior, as $N$ grows, of the largest minimal pairwise distance of $N$ points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz $s$-energy configurations and determine the $s$-th root asymptotic behavior (as $sto infty)$ of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in ${rm {bf R}}^d$ of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal $s$-energy for large $s$ is different for different subsequences of the cardinalities of the configurations.
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.
Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $Nto infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e, for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d.$ As a consequence, we immediately get (thanks to the Poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $exp(-alpha|x-y|^2)$ on $mathbb{R}^p,$ we obtain lower bounds for the energy of infinite configurations having a prescribed density.
Observations suggest that configurations of points on a sphere that are stable with respect to a Riesz potential distribute points uniformly over the sphere. Further, these stable configurations have a local structure that is largely hexagonal. Minimal configurations differ from stable configurations in the arrangement of defects within the hexagonal structure. This paper reports the asymptotic difference between the average energy of stable states and the lowest reported energies. We use this to infer the energy scale at which defects in the hexagonal structure are manifest. We report results for the Riesz potentials for s=0, s=1, s=2 and s=3. Additionally we compare existing theory for the asymptotic expansion in N of the minimal $N$-point energy with experimental results. We report a case of two distinct stable states that have the same Voronoi structure. Finally, we report the observed growth of the number of stable states as a function of N.
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