## Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations

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##### Date

2013-06-10##### MFO Scientific Program

OWLF 2013##### Series

Oberwolfach Preprints;2013,13##### Author

Bondarenko, A. V.

Hardin, Douglas P.

Saff, Edward B.

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Show full item record##### OWP-2013-13

##### Abstract

For $N$-point best-packing configurations $\omega_N$ on a compact metric
space $(A, \rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\rho_N , A)$, 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
$\gamma(\omega_N , A) ≤ 1$ and this bound can be attained even for the sphere. In the particular
case when $N = 5$ on $S^1$
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 $s \to \infty$) of
5-point $s$-energy minimizing configurations. Moreover, $\gamma(\omega^*_5, S^2) = 1$.