IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4614-1129-1_2.html

Bounds for Local Density of Sphere Packings and the Kepler Conjecture

In: The Kepler Conjecture

Author

Listed:
  • J. C. Lagarias

    (AT&T Labs - Research)

Abstract

This paper formalizes the local density inequality approach to getting upper bounds for sphere packing densities in ℝn. This approach was first suggested by L. Fejes Tóth in 1953 as a method to prove the Kepler conjecture that the densest packing of unit spheres in ℝ3 has density $$\pi / \sqrt{18}$$ , which is attained by the “cannonball packing.” Local density inequalities give upper bounds for the sphere packing density formulated as an optimization problem of a nonlinear function over a compact set in a finite-dimensional Euclidean space. The approaches of Fejes Tóth, of Hsiang, and of Hales to the Kepler conjecture are each based on (different) local density inequalities. Recently Hales, together with Ferguson, has presented extensive details carrying out a modified version of the Hales approach to prove the Kepler conjecture. We describe the particular local density inequality underlying the Hales and Ferguson approach to prove Kepler’s conjecture and sketch some features of their proof.

Suggested Citation

  • J. C. Lagarias, 2011. "Bounds for Local Density of Sphere Packings and the Kepler Conjecture," Springer Books, in: Jeffrey C. Lagarias (ed.), The Kepler Conjecture, chapter 2, pages 27-57, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-1129-1_2
    DOI: 10.1007/978-1-4614-1129-1_2
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-1-4614-1129-1_2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.