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

## TL;DR

This paper introduces a new local invariant for sphere packing in three dimensions, providing a simplified proof of Kepler's conjecture and related principles by characterizing optimal packings as f.c.c. or h.c.p.

## Contribution

It defines a novel local density invariant using only a single layer of spheres, leading to simplified proofs of Kepler's conjecture and crystal formation principles.

## Key findings

- Optimal local packings are f.c.c. or h.c.p.
- New local invariant characterizes dense packings uniquely.
- Simplified proofs of Kepler's conjecture and crystal formation principles.

## 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].