On the Densities of Certain Lattice Packings by Parallelepipeds

Given any (non-degenerate) n-dimensional lattice L, let (L) denote the supremum of the numbers such that there exists a lattice packing Q + L of density where Q is some o-symmetric parallelepiped with faces parallel to the coordinate axes. Many efforts have been made to determine or estimate the minimal such density _n taken over all n-dimensional lattices. It is known that . Here we investigate a sequence of lattices L_n which are known to minimize the function (L) in dimensions n 3 and are likely to provide the minima _n = (L_n) in certain higher dimensions. We establish the inequality (L_n) n^–n/2 which supports the conjecture that lim sup_n (_n)^{1/(n log n)} is positive.