Sphere Packings, IV. Detailed Bounds

  • Thomas C. Hales


This paper is the fourth in a series of six papers devoted to the proof of theKepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In a previous paper in this series, a continuous function f on a compact spacewas 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. The function f can be expressed as a sum of terms, indexed by regions on a unit sphere. In this paper detailed estimates of the terms corresponding to general regions are developed. These results form the technical heart of the proof of the Kepler conjecture, by giving detailed bounds on the function f. The results rely on long computer calculations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ha6.
    T. C. Hales, Sphere packings, I, Discrete Comput. Geom. 17 (1997), 1–51.MathSciNetCrossRefGoogle Scholar

Copyright information

© T.C. Hales 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

Personalised recommendations