Abstract. Given a configuration of (solid) spheres in 3 dimensions, the density is the expected number of spheres that contain a random point (assuming this expectation is defined). Equivalently, it is $p_1 + p_2 + \ldots$, in which $p_i$ is the probability that a random point is contained in at least $i$ spheres. Among the packings, the density is maximized by the centers in the FCC lattice, and among the coverings, the density is minimized by the centers in the BCC lattice. We introduce the 1-parameter diagonal family of lattices---which includes FCC and BCC---and give closed-form expressions for the densities of all its packings and coverings.

To get a compromise between packing and covering, we call $p_1 - p_2 - p_3 - \ldots$ the soft density, and we show that among the lattices in the diagonal family, it is maximized by the FCC lattice, but the BCC lattice is almost as good.

The first part is joint work with Michael Kerber and the second is joint work with Mabel Iglesias-Ham.