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arXiv · 1503.04571

The packing density of the $n$-dimensional cross-polytope

G. Fejes Tóth·F. Fodor·V. Vígh
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Abstract

The packing density of the regular cross-polytope in Euclidean $n$-space is unknown except in dimensions $2$ and $4$ where it is 1. The only non-trivial upper bound is due to Gravel, Elser, and Kallus (2011) who proved that for $n=3$ the packing density of the regular octahedron is at most $1-1.4\ldots\times 10^{-12}$. In this paper, we prove upper bounds for the packing density of the $n$-dimensional regular cross-polytope in the case that $n\geq 7$. We use a modification of Blichfeldt's method due to G. Fejes T\'oth and W. Kuperberg (1993).

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G. Fejes Tóth, F. Fodor, V. Vígh. 2015-03-16. The packing density of the $n$-dimensional cross-polytope. https://doi.org/10.1007/s00454-015-9699-5

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