A new local invariant and simpler proof of Kepler's conjecture and the least action principle on the crystalformation of dense type

Abstract

A new locally averaged density for sphere packing in R^3 is defined by a proper combination of the local cell (Voronoi cell) and Delaunay decompositions (\S 1.2.2), using only a single layer of surrounding spheres. Local packings attaining the optimal estimate of such a local invariant must be either the f.c.c. or h.c.p. local packings (Theorem I). The main purpose of this paper is to provide a clean-cut proof of this strong uniqueness result via geometric invariant theory. This result also leads to simple proofs of Kepler's conjecture on sphere packing, least action principle of crystal formation of dense type, and optimal packings with containers (Theorems II-IV). This work provides a much simplified alternative to the author's previous work on Kepler's conjecture and least action principle of crystal formation of dense type which involved a local invariant defined by double layer of surrounding spheres [Hsi].