Bounds for Local Density of Sphere Packings and the Kepler Conjecture

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 π/\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.