## Three-point bounds for energy minimization

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- by Henry Cohn and Jeechul Woo
- J. Amer. Math. Soc.
**25**(2012), 929-958 - DOI: https://doi.org/10.1090/S0894-0347-2012-00737-1
- Published electronically: May 1, 2012
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## 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.## References

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## Bibliographic Information

**Henry Cohn**- Affiliation: Microsoft Research New England, One Memorial Drive, Cambridge, Massachuetts 02142
- MR Author ID: 606578
- ORCID: 0000-0001-9261-4656
- Email: cohn@microsoft.com
**Jeechul Woo**- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- Email: woo@math.harvard.edu
- 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.
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**25**(2012), 929-958 - MSC (2010): Primary 05B40, 52A40, 52C17; Secondary 90C22, 82B05
- DOI: https://doi.org/10.1090/S0894-0347-2012-00737-1
- MathSciNet review: 2947943