Sphere Packings, III. Extremal Cases
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.
Unable to display preview. Download preview PDF.
- Ha16.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.)