A Bound on the Ratio between the Packing and Covering Densities of a Convex Body

It is shown that if K is a compact convex set which is centrally symmetric and has a nonempty 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 1/4. It is likely this bound can be improved, though not beyond approximately 1/2. Received October 8, 1998, and in revised form December 30, 1998.