Bounds for Local Density of Sphere Packings and the Kepler Conjecture
Jeffrey C. Lagarias
Abstract
This paper describes the local density inequality approach to getting upper bounds for sphere packing densities in Rn. This approach was first suggested by L. Fejes-Toth in 1956 to prove the Kepler conjecture that the densest sphere packing in R3 is the "cannonball packing". The approaches of L. Fejes-Toth, W.-Y. Hsiang, and T. Hales to the Kepler conjecture are each based on (different) local density inequalities. Recently T. Hales and S. P. Ferguson have presented extensive details carrying out a modified version of the otrignal Hales approach to prove the Kepler conjecture. We describe the particular local density inequality undelying the Hales and Ferguson approach and sketch some features of their proof.
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