Book chapter
An Approach to the Dodecahedral Conjecture Based on Bounds for Spherical Codes
Discrete Geometry and Optimization, pp.33-44
Fields Institute Communications, Springer International Publishing
05/17/2013
DOI: 10.1007/978-3-319-00200-2_3
Abstract
The dodecahedral conjecture states that in a packing of unit spheres in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$${\mathfrak{R}}^{3}$$
\end{document}, 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.
Details
- Title: Subtitle
- An Approach to the Dodecahedral Conjecture Based on Bounds for Spherical Codes
- Creators
- Kurt M. Anstreicher - University of Iowa
- Resource Type
- Book chapter
- Publication Details
- Discrete Geometry and Optimization, pp.33-44
- Publisher
- Springer International Publishing; Heidelberg
- Series
- Fields Institute Communications
- DOI
- 10.1007/978-3-319-00200-2_3
- eISSN
- 2194-1564
- ISSN
- 1069-5265
- Language
- English
- Date published
- 05/17/2013
- Academic Unit
- Industrial and Systems Engineering; Computer Science; Business Analytics
- Record Identifier
- 9984380715302771
Metrics
1 Record Views