Edge close ball packings (Q5939540)

Consider a packing of unit balls with center \(\{o_i\}\). The Dirichlet-Voronoi cell associated with \(o_i\) consists of the points in space which are closer to \(o_i\) than any other center. An edge-line is the intersection of the planes of two intersecting faces of some Dirichlet-Voronoi cell, and the edge-closeness of a packing is the maximal distance between an edge-line (of a Dirichlet-Voronoi cell) and the center of (one of the) corresponding unit balls. The author shows that the unique packing with minimum edge-closeness is generated by the space-centered cubic lattice. This is the main result of this substantial 12 page paper which contains 3 Propositions, 2 Theorems, 6 Lemmas, 1 Remark, 1 Corollary and 16 Bibliographic entries, and reveals significant results having to do with the structure of polyhedra, Dirichlet-Voronoi cells, tilings and more.