TY - JOUR

T1 - On the packing densities of superballs and other bodies

AU - Elkies, N. D.

AU - Odlyzko, A. M.

AU - Rush, J. A.

PY - 1991/12/1

Y1 - 1991/12/1

N2 - 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 classical lσ-ball, {Mathematical expression} 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 large n, where {Mathematical expression} 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 {Mathematical expression} where f 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.

AB - 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 classical lσ-ball, {Mathematical expression} 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 large n, where {Mathematical expression} 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 {Mathematical expression} where f 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.

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U2 - 10.1007/BF01232282

DO - 10.1007/BF01232282

M3 - Article

AN - SCOPUS:33746373749

SN - 0020-9910

VL - 105

SP - 613

EP - 639

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -