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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Universally optimal distribution of points on spheres


Authors: Henry Cohn and Abhinav Kumar
Journal: J. Amer. Math. Soc. 20 (2007), 99-148
MSC (2000): Primary 52A40, 52C17; Secondary 41A05
Published electronically: September 5, 2006
MathSciNet review: 2257398
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Abstract: We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are $ m$ distances between distinct points in it and it is a spherical $ (2m-1)$-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the $ E_8$ and Leech lattices. We also prove the same result for the vertices of the $ 600$-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.


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Additional Information

Henry Cohn
Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
Email: cohn@microsoft.com

Abhinav Kumar
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
Email: abhinav@math.harvard.edu, abhinavk@microsoft.com

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00546-7
PII: S 0894-0347(06)00546-7
Keywords: Potential energy minimization, spherical codes, spherical designs, sphere packing
Received by editor(s): November 1, 2004
Published electronically: September 5, 2006
Additional Notes: The second author was supported by a summer internship in the Theory Group at Microsoft Research and a Putnam Fellowship at Harvard University.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.