A lower bound on packing density. (English) Zbl 0659.10033

Rush, J. A.

Invent. Math. 98, No. 3, 499-509 (1989).

If \(H\subseteq {\mathbb{R}}^ n\) is a bounded, convex body which is symmetric through each of the coordinate hyperplanes, then there exist codes which give rise, via Construction A of Leech and Sloane to lattice-packings of H whose density \(\Delta\) satisfies the logarithmic Minkowski-Hlawka bound, \(\liminf_{n\to \infty}\log_ 2^ n\sqrt{\Delta}\geq -1.\) This follows as a corollary of our main result, Theorem 9, a general way of obtaining lower bounds on the lattice-packing densities of various bodies. Unfortunately, when n is at all large, it is computationally prohibitive (although theoretically possible) to exhibit the arrangements explicitly.

Reviewer: J.A.Rush

MSC:

11H31	Lattice packing and covering (number-theoretic aspects)
52C17	Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11H06	Lattices and convex bodies (number-theoretic aspects)

Keywords:

convex body; construction A; logarithmic Minkowski-Hlawka bound; lower bounds; lattice-packing densities

References:

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