On the packing densities of superballs and other bodies

Abstract

A method of obtaining improvements to the Minkowski-Hlawka bound on the lattice-packing density for many convex bodies symmetrical through the coordinate hyperplanes, described by Rush [18], is generalized so that centrally symmetric convex bodies can be treated as well. The lower bounds which arise are very good. The technique is applied to various shapes, including the classicall σ-ball, $$\left\{ {x \in R^n :|x_1 |^\sigma + |x_2 |^\sigma + ... + |x_n |^\sigma \leqq 1} \right\},$$ for σ≧1. This generalizes the earlier work of Rush and Sloane [17] in which σ was required to be an integer. The superball above can be lattice packed to a density of(b/2) n+0(1) for largen, where $$b = \mathop {\sup }\limits_{t > 0} \frac{{\int\limits_{x = - \infty }^\infty {e^{ - | tx |^\sigma d x} } }}{{\sum\limits_{k = - \infty }^\infty {e^{ - | tx |^\sigma } } }}.$$ This is as good as the Minkowski-Hlawka bound for 1≦σ≦2, and better for σ>2. An analogous density bound is established for superballs of the shape $$\left\{ {x \in R^n :f(x_1 ,...,x_k )^\sigma + f(x_{k + 1} ,...,x_{2k} )^\sigma + ... + f(x_{n - k + 1} ,...,x_n )^\sigma \leqq 1} \right\}, k|n,$$ wheref is the Minkowski distance function associated with a bounded, convex, centrally symmetric,k-dimensional body. Finally, we consider generalized superballs for which the defining inequality need not even be homogeneous. For these bodies as well, it is often possible to improve on the Minkowski-Hlawka bound.