Geodesic
Belt Distance Between Facets of Space-Filling Zonotopes
Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 381-394

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To every $d$-dimensional polytope $P$ with centrally symmetric facets, one can assign a “subway map” such that every line of this “subway” contains exactly the facets parallel to one of the ridges of $P$. The belt diameter of $P$ is the maximum number of subway lines one needs to use to get from one facet to another. We prove that the belt diameter of a $d$-dimensional space-filling zonotope does not exceed $\lceil\log_2(4/5)d\rceil$.
Keywords: zonotope, polytope, belt diameter, tiling, Dirichlet–Voronoi polytope, canonical scaling of a tiling.
Mots-clés : parallelohedron, Voronoi's conjecture 
@article{MZM_2012_92_3_a5,
     author = {A. I. Garber},
     title = {Belt {Distance} {Between} {Facets} of {Space-Filling} {Zonotopes}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {381--394},
     publisher = {mathdoc},
     volume = {92},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a5/}
}
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A. I. Garber. Belt Distance Between Facets of Space-Filling Zonotopes. Matematičeskie zametki, Tome 92 (2012) no. 3, pp. 381-394. http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a5/