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Minimal $ N$-point diameters and $ f$-best-packing constants in $ \mathbb{R}^d$


Authors: A. V. Bondarenko, D. P. Hardin and E. B. Saff
Journal: Proc. Amer. Math. Soc. 142 (2014), 981-988
MSC (2010): Primary 52C17
Published electronically: December 11, 2013
MathSciNet review: 3148532
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Abstract: In terms of the minimal $ N$-point diameter $ D_d(N)$ for $ \mathbb{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(\Vert x-y\Vert)\, :\, x,y\in \mathbb{R}^d\}$, where the minimum is taken over point sets of cardinality $ N.$ We also show that

$\displaystyle N^{1/d}\Delta _d^{-1/d}-2\le D_d(N)\le N^{1/d}\Delta _d^{-1/d}, \quad N\ge 2,$

where $ \Delta _d$ is the maximal sphere packing density in $ \mathbb{R}^d$. Further, we provide asymptotic estimates for the $ f$-best-packing constants as $ N\to \infty $.

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Additional Information

A. V. Bondarenko
Affiliation: Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain — and — Department of Mathematical Analysis, National Taras Shevchenko University, str. Volodymyrska 64, Kyiv, 01033, Ukraine
Address at time of publication: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Email: andriybond@gmail.com

D. P. Hardin
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: Doug.Hardin@Vanderbilt.Edu

E. B. Saff
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: Edward.B.Saff@Vanderbilt.Edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11657-6
Keywords: Best-packing, optimal configurations, minimal $N$-point diameter, maximal sphere packing density
Received by editor(s): November 3, 2010
Received by editor(s) in revised form: November 1, 2011, and April 17, 2012
Published electronically: December 11, 2013
Additional Notes: The research of the first author was conducted while visiting the Center for Constructive Approximation in the Department of Mathematics, Vanderbilt University
The research of all the authors was supported, in part, by the U.S. National Science Foundation under grant DMS-0808093
Communicated by: Jim Haglund
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.