This article is cited in 2 papers

Belt Distance Between Facets of Space-Filling Zonotopes

A. I. Garber<sup>ab

a</sup> M. V. Lomonosov Moscow State University
^b P. G. Demidov Yaroslavl State University

Abstract: 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, parallelohedron, polytope, belt diameter, Voronoi's conjecture, tiling, Dirichlet–Voronoi polytope, canonical scaling of a tiling.

UDC: 514

Received: 16.12.2010
Revised: 12.02.2011

DOI: 10.4213/mzm9059

 English version: Mathematical Notes, 2012, 92:3, 345–355

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