Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Discrete & Computational Geometry
  3. Article
Sphere Packing, III. Extremal Cases
Download PDF
Download PDF
  • Published: 27 February 2006

Sphere Packing, III. Extremal Cases

  • Thomas C. Hales1 

Discrete & Computational Geometry volume 36, pages 71–110 (2006)Cite this article

  • 232 Accesses

  • 12 Citations

  • Metrics details

Abstract

This paper is the third in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In the previous paper in this series, a continuous function f on a compact space was defined, certain points in the domain were conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture was established. This paper shows that those points are indeed local maxima. Various approximations to f are developed, that will be used in subsequent papers to bound the value of the function f. The function f can be expressed as a sum of terms, indexed by regions on a unit sphere. Detailed estimates of the terms corresponding to triangular and quadrilateral regions are developed.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15217, USA

    Thomas C. Hales

Authors
  1. Thomas C. Hales
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas C. Hales.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hales, T. Sphere Packing, III. Extremal Cases. Discrete Comput Geom 36, 71–110 (2006). https://doi.org/10.1007/s00454-005-1212-0

Download citation

  • Received: 11 November 1998

  • Revised: 25 July 2005

  • Published: 27 February 2006

  • Issue Date: July 2006

  • DOI: https://doi.org/10.1007/s00454-005-1212-0

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Continuous Function
  • Computational Mathematic
  • Extremal Case
  • Local Maximum
  • Unit Sphere
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature