Inventiones mathematicae

, Volume 105, Issue 1, pp 613–639

# On the packing densities of superballs and other bodies

• N. D. Elkies
• A. M. Odlyzko
• J. A. Rush
Article

## Summary

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 classicallσ-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.

## Keywords

Lower Bound Distance Function Packing Density Convex Body Symmetric Convex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• N. D. Elkies
• 1
• A. M. Odlyzko
• 2
• J. A. Rush
• 3
1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
2. 2.AT & T Bell LaboratoriesMurray HillUSA
3. 3.Department of MathematicsUniversity of EdinburghEdinburghScotland, UK