Locally dense finite lattice packings of spheres (Q1180737)

The author considers lattices \(L\) in \(\mathbb{E}^ 3\) which are packing lattices of the unit ball. Let \(h(L)\) denote the maximum distance of consecutive parallel 2-dimensional ``layers'' of \(L\). The first result supports the conjecture that \(h(L)\geq (8/3)^{1/2}\) for each such lattice \(L\). It is pointed out that there are at least two different such lattices for which equality holds. In the second result is it shown that the Dirichlet-Voronoi cell of such a lattice \(L\) lies strictly between the two lattice layers next to the origin and which have mutual distance \(2h(L)\).