Isoperimetric inequalities and the dodecahedral conjecture

The dodecahedrad conjecture, posed more than 50 years ago, says that the volume of any Voronoi polyhedron of a unit sphere packing in E-3 is at least as large as the volume of a regular dodecahedron of inradius 1. In this paper we show how the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi laces of unit sphere packings.