Lower bound on the translative covering density of octahedra. (English) Zbl 07844819
A translative covering of \(\mathbb R^3\) by the octahedron \(C_3=\mathrm{conv}\{(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)\}\) is a family of translates \(\{C_3+t_i\}_{i=1}^\infty\) such that \(\bigcup_{i=1}^\infty C_3+t_i=\mathbb R^3\). Its (lower) density is \(\liminf_{l \to \infty}\frac{\sum_{i=1}^\infty\mathrm{vol}((C_3+t_i) \cap [-l,l]^3)}{\mathrm{vol}([-l,l]^3)}\). It is shown that the density of every such covering is bounded from below by \(1+4 \cdot 6^{-10} > 1+6.6 \cdot 10^{-8}\).
Reviewer: Christian Richter (Jena)
MSC:
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
| 05C12 | Distance in graphs |
| 52B10 | Three-dimensional polytopes |
| 52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |
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