A Density Bound for Efficient Packings of 3-space with Centrally Symmetric Convex Bodies

An Improvement of an Inequality Linking Packing and Covering Densities in 3-space

Abstract
It is shown that if K is a compact convex set which is centrally symmetric and has a non-empty interior, then the density of the tightest lattice packing with
copies of K in Euclidean 3-space divided by the density of the thinnest lattice covering of Euclidean 3-space with copies of K is greater than or equal to 1/3.
This improves the previous bound in [Smith00] of 1/4. It is possible this bound will be improved in the future, though not beyond approximately 1/2.