Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotics of best-packing on rectifiable sets


Authors: S. V. Borodachov, D. P. Hardin and E. B. Saff
Journal: Proc. Amer. Math. Soc. 135 (2007), 2369-2380
MSC (2000): Primary 11K41, 70F10, 28A78; Secondary 78A30, 52A40
Published electronically: April 10, 2007
MathSciNet review: 2302558
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $ s\to \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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11K41, 70F10, 28A78, 78A30, 52A40

Retrieve articles in all journals with MSC (2000): 11K41, 70F10, 28A78, 78A30, 52A40


Additional Information

S. V. Borodachov
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332
Email: borodasv@math.gatech.edu

D. P. Hardin
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: doug.hardin@vanderbilt.edu

E. B. Saff
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: Edward.B.Saff@Vanderbilt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08975-7
PII: S 0002-9939(07)08975-7
Keywords: Best-packing points, sphere packing, rectifiable set, Thomson problem, packing measure, minimal discrete Riesz energy, hard spheres problem
Received by editor(s): April 19, 2006
Published electronically: April 10, 2007
Additional Notes: The research of the second author was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0532154
The research of the third author was supported, in part, by the U. S. National Science Foundation under grant DMS-0532154.
Communicated by: David Preiss
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.