A new bound on the local density of sphere packings (Q1312189)

Continuing a number of results on and improvements of upper bounds on the density of (nonoverlapping) packings of unit spheres in 3-space, the author shows that such a density is at most \(0,773055\dots\); for showing this, he uses the Voronoi polyhedra of such packings. (Such a Voronoi polyhedron \(P\) is the set of points that lie closer to the midpoint of the corresponding sphere \(S\) than to the center of another sphere from the packing. Thus, the volume ratio \(V(S)\over V(P)\) is a local measure of density, and any upper bound on the local densities is an upper bound on the density of the packing as a whole.) Namely, the author shows that a Voronoi polyhedron defined by such a sphere packing must have volume at least \(5,41848\dots\); by cutting the Voronoi polyhedron into cones (one for each of its facets), a lower bound on the cone volumes is established as a function of the respective solid angle. The sum of all the cone volume bounds is minimized when there are 13 facets, each of solid angle \(4\pi\over 13\).