| Abstract: | A typical 3-dimensional (in short '3D') Voronoi cell of a 3Dlattice has six families of parallel edges. We call any six representants of these six families the generating edges of the Voronoi cell. The sum s of lengths of generating edges of a Voronoi-cell of a lattice unit sphere packing in the 3-dimensional Euclidean space is
a special case of intrinsic 1-volumes of 3Dzonotopes with inradius 1 which are investigated accurately in B]. However, the minimum of this value is unknown even in
this special case. As the regular rhombic dodecahedron shows optimal properties in many similar problems, it was reasonable
to conjecture that it also has the minimal s value. In this note we present a construction of a lattice unit ball packing whose Voronoi cell possesses an intrinsic 1-volume
strictly less than the one of the proper regular rhombic dodecahedron, hence providing a smaller upper bound for s than it was conjectured. A further issue of the note is a formula for edge-lengths of Voronoi cells of lattice unit ball
packings that can be used efficiently in similar calculations.
This revised version was published online in August 2006 with corrections to the Cover Date. |
