Summary
If ℋ\( \subseteq\) ℝnis 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 Δ satisfies the logarithmic Minkowski-Hlawka bound, lim\(\inf _{n \to \infty } \log _2 \sqrt[n]{\Delta } \geqq - 1\). 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, whenn is at all large, it is computationally prohibitive (although theoretically possible) to exhibit the arrangements explicitly.
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Preprints were entitled On Hilbert's eighteenth problem: Packing.
The final drafts of this paper were written with the support of the Seggie-Brown Research Fellowship, for which the author is grateful.
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Rush, J.A. A lower bound on packing density. Invent Math 98, 499–509 (1989). https://doi.org/10.1007/BF01393834
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DOI: https://doi.org/10.1007/BF01393834