Inequalities between lattice packing and covering densities of centrally symmetric plane convex bodies

Given a family C of plane convex bodies, let Omega (C) be the set of all pairs (x, y) with the property that there exists K is an element of C such that theta (K) = x and delta (K) = y, where theta (K) and delta (K) denote the densities of the thinnest covering and the densest packing of the plane with copies of K, respectively. The set R-L(C) is defined analogously, with the difference that we restrict our attention to lattice packings and coverings. We prove that, for every centrally symmetric plane convex body K, 1 - delta (L) (K) less than or equal to theta (L)(K) - 1 less than or equal to 1.25 root1 - delta (L)(K) and give an exact analytic description of Omega (L)(P-8) where P-8 is the family of all centrally symmetric octagons. This allows us to show that the above inequalities are asymptotically tight.