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Three-point bounds for energy minimization

Authors: Henry Cohn and Jeechul Woo
Journal: J. Amer. Math. Soc. 25 (2012), 929-958
MSC (2010): Primary 05B40, 52A40, 52C17; Secondary 90C22, 82B05
Published electronically: May 1, 2012
Supplement 1: data1.txt
Supplement 2: data2.txt
Supplement 3: data3.txt
Supplement 4: data4.txt
Supplement 5: data5.txt
Supplement 6: definitions.txt
Supplement 7: optimal.txt
Supplement 8: unique.txt
MathSciNet review: 2947943
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Abstract | References | Similar Articles | Additional Information

Abstract: Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for the potential energy of particles interacting via a pair potential function. We show that our bounds are sharp for seven points in $ \mathbb{R}\mathbb{P}^2$. Specifically, we prove that the seven lines connecting opposite vertices of a cube and of its dual octahedron are universally optimal. (In other words, among all configurations of seven lines through the origin, this one minimizes energy for all potential functions that are completely monotonic functions of squared chordal distance.) This configuration is the only known universal optimum that is not distance regular, and the last remaining universal optimum in $ \mathbb{R}\mathbb{P}^2$. We also give a new derivation of semidefinite programming bounds and present several surprising conjectures about them.

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Additional Information

Henry Cohn
Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachuetts 02142

Jeechul Woo
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): March 2, 2011
Received by editor(s) in revised form: February 24, 2012
Published electronically: May 1, 2012
Additional Notes: The second author was supported in part by an internship at Microsoft Research New England and by a Samsung Scholarship.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.