Bounds for Local Density of Sphere Packings and the Kepler Conjecture

  • J. C. Lagarias


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. C.
    H. Cohn, New Bounds on Sphere Packings, Thesis, Harvard University, April 2000.Google Scholar
  2. CS.
    J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Codes, Third edition, Springer-Verlag: New York, 1999.Google Scholar
  3. Dod.
    T. C. Hales and S. McLaughlin, A proof of the dodecahedral conjecture, eprint: arXiv math.MG/9811079.Google Scholar
  4. FT1.
    L. Fejes T´oth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer-Verlag: Berlin, 1953. (Second edition, 1972.)Google Scholar
  5. FT2.
    L. Fejes T´oth, Regular Figures, MacMillan: New York, 1964.Google Scholar
  6. FH.
    S. P. Ferguson and T. C. Hales, A formulation of the Kepler conjecture, eprint: arXiv math.MG/9811072.Google Scholar
  7. Ha1.
    T. C. Hales, The sphere packing problem, J. Comput. Appl. Math. 44 (1992), 41–76.MathSciNetCrossRefGoogle Scholar
  8. Ha2.
    T. C. Hales, Remarks on the density of sphere packings in three dimensions, Combinatorica 13(2) (1993), 181–197.MathSciNetCrossRefGoogle Scholar
  9. Ha3.
    T. C. Hales, The status of the Kepler conjecture, Math. Intelligencer 16(3) (1994), 47–58.MathSciNetCrossRefGoogle Scholar
  10. Ha4.
    T. C. Hales, Cannonballs and honeycombs, Notices Amer. Math. Soc. 47(4) (2000), 440–449.Google Scholar
  11. Hi.
    D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437–479. Reprinted in Mathematical Developments Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics, XXVIII, American Mathematical Society: Providence, RI, 1976.Google Scholar
  12. Hs1.
    W.-Y. Hsiang,Onthe sphere problem andKepler’s conjecture, Internat. J. Math. 4(5) (1993), 739–831. (MR 95g: 52032.)Google Scholar
  13. Hs2.
    W.-Y. Hsiang, A rejoinder to Hales’ article, Math. Intelligencer 17(1) (1995), 35–42.MathSciNetCrossRefGoogle Scholar
  14. KC.
    T. C. Hales, The Kepler conjecture, eprint: arXiv math.MG/9811078.Google Scholar
  15. KC0.
    T. C. Hales, An overview of the Kepler conjecture, eprint: arXiv math.MG/9811071.Google Scholar
  16. L.
    J. C. Lagarias, Notes on the Hales approach to the Kepler conjecture, manuscript, May 1999.Google Scholar
  17. O.
    J. Oesterl´e, Densit´e maximale des empilements de sph`eres en dimension 3 [d’apr`es Thomas C. Hales et Samuel P. Ferguson], S´eminaire Bourbaki, Vol. 1998/99. Aster´isque 266 (2000), Exp. No. 863, 405–413.Google Scholar
  18. R1.
    C. A. Rogers, The packing of equal spheres, Proc. London Math. Soc. 8 (1958), 609–620.MathSciNetCrossRefGoogle Scholar
  19. R2.
    C. A. Rogers, Packing and Covering, Cambridge University Press: Cambridge, 1964.Google Scholar
  20. SP-I.
    T. C. Hales, Sphere packings, I, Discrete Comput. Geom. 17 (1997), 1–51, eprint: arXiv math.MG/9811073.Google Scholar
  21. SP-II.
    T. C. Hales, Sphere packings, II, Discrete Comput. Geom. 18 (1997), 135–149, eprint: arXiv math.MG/9811074.MathSciNetCrossRefGoogle Scholar
  22. SP-III.
    T. C. Hales, Sphere packings, III, eprint: arXiv math.MG/9811075.Google Scholar
  23. SP-IV.
    T. C. Hales, Sphere packings, IV, eprint: arXiv math.MG/9811076.Google Scholar
  24. SP-V.
    S. P. Ferguson, Sphere packings, V, Thesis, University of Michigan, 1997, eprint: arXivmath.MG/9811077.Google Scholar
  25. Z.
    C. Zong, Sphere Packings, Springer-Verlag: New York, 1999.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.AT&T Labs - ResearchFlorham ParkUSA

Personalised recommendations