A proof of the dodecahedral conjecture

TL;DR

This paper proves the dodecahedral conjecture, establishing a lower bound on Voronoi polyhedron volume in equal sphere packings, using a methodology similar to that of the Kepler conjecture proof.

Contribution

It provides the first rigorous proof of the dodecahedral conjecture, extending techniques from the Kepler conjecture to Voronoi polyhedra.

Findings

Confirmed the volume bound for Voronoi polyhedra in sphere packings.

Established a new geometric inequality related to dodecahedral volume.

Extended proof techniques from the Kepler conjecture to this conjecture.

Abstract

The dodecahedral conjecture states that the volume of the Voronoi polyhedron of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. The authors prove the conjecture following the methodology of the proof the Kepler conjecture. (See math.MG/9811071.)