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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Universally optimal distribution of points on spheres
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by Henry Cohn and Abhinav Kumar PDF
J. Amer. Math. Soc. 20 (2007), 99-148 Request permission

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.
References
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Additional Information
  • Henry Cohn
  • Affiliation: Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
  • MR Author ID: 606578
  • ORCID: 0000-0001-9261-4656
  • 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
  • MR Author ID: 694441
  • Email: abhinav@math.harvard.edu, abhinavk@microsoft.com
  • 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.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 20 (2007), 99-148
  • MSC (2000): Primary 52A40, 52C17; Secondary 41A05
  • DOI: https://doi.org/10.1090/S0894-0347-06-00546-7
  • MathSciNet review: 2257398