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An Approach to the Dodecahedral Conjecture Based on Bounds for Spherical Codes

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Part of the Fields Institute Communications book series (FIC, volume 69)

Abstract

The dodecahedral conjecture states that in a packing of unit spheres in \({\mathfrak{R}}^{3}\), the Voronoi cell of minimum possible volume is a regular dodecahedron with inradius one. The conjecture was first stated by L. Fejes Tóth in 1943, and was finally proved by Hales and McLaughlin over 50 years later using techniques developed by Hales for his proof of the Kepler conjecture. In 1964, Fejes Tóth described an approach that would lead to a complete proof of the dodecahedral conjecture if a key inequality were established. We describe a connection between the key inequality required to complete Fejes Tóth’s proof and bounds for spherical codes and show how recently developed strengthened bounds for spherical codes may make it possible to complete Fejes Tóth’s proof.

Key words

Dodecahedral conjecture Kepler conjecture Spherical codes Delsarte bound Semidefinite programming 

Subject Classifications

52C17 90C22 90C26 

Notes

Acknowledgements

I would like to thank Tibor Csendes for providing an English translation of [5], and Frank Vallentin for independently verifying the computations based on SDP(m). I am also grateful to two anonymous referees for their careful readings of the paper and valuable suggestions to improve it.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA

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