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.
Received November 11, 1998, and in revised form September 12, 2003, and July 25, 2005. Online publication February 27, 2006.
The original version of this chapter was revised. An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4614-1129-1_12
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References
T. C. Hales, The sphere packing problem, J. Comput. Appl. Math. 44 (1992), 41–76.
T. C. Hales, Sphere packings, I, Discrete Comput. Geom. 17 (1997), 1–51.
T. C. Hales, Sphere packings, II, Discrete Comput. Geom. 18 (1997), 135–149.
T. C. Hales, Some algorithms arising in the proof of theKepler conjecture, Discrete and Computational Geometry, Algorithms and Combinatorics, vol. 25, Springer-Verlag, Berlin, 2003, pp. 489–507.
T. C. Hales, Computer Resources for the Kepler Conjecture, http://annals.math.princeton.edu/keplerconjecture/. (The source code, inequalities, and other computer data relating to the solution are also found at http://xxx.lanl.gov/abs/math/ 9811078v1.)
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© 2011 T.C. Hales
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Hales, T.C. (2011). Sphere Packings, III. Extremal Cases. In: Lagarias, J.C. (eds) The Kepler Conjecture. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1129-1_5
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DOI: https://doi.org/10.1007/978-1-4614-1129-1_5
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