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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - A. I. Garber
TI  - Belt Distance Between Facets of Space-Filling Zonotopes
JO  - Matematičeskie zametki
PY  - 2012
SP  - 381
EP  - 394
VL  - 92
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a5/
LA  - ru
ID  - MZM_2012_92_3_a5
ER  - 
%0 Journal Article
%A A. I. Garber
%T Belt Distance Between Facets of Space-Filling Zonotopes
%J Matematičeskie zametki
%D 2012
%P 381-394
%V 92
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2012_92_3_a5/
%G ru
%F MZM_2012_92_3_a5
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/

[1] G. Voronoï, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloèdres primitifs”, J. für Math., 136 (1909), 67–178 | Zbl

[2] H. Minkowski, “Allgemeine Lehrsätze über die convexen Polyeder”, Gött. Nachr., 1897, 198–219 | Zbl

[3] P. McMullen, “Convex bodies which tile space by translation”, Mathematika, 27:1 (1980), 113–121 | DOI | MR | Zbl

[4] B. A. Venkov, “Ob odnom klasse evklidovykh mnogogrannikov”, Vest. Leningradsk. un-ta. Ser. matem., fiz., khim., 9:2 (1954), 11–31 | MR

[5] O.K. Zhitomirskii, “Verschärfung eines Satzes von Voronoi”, Zh. Leningradskogo. matem. ob-va, 2 (1929), 131–151

[6] R. M. Erdahl, “Zonotopes, dicings, and Voronoi's conjecture on parallelohedra”, European J. Combin., 20:6 (1999), 527–549 | DOI | MR | Zbl

[7] A. Ordine, Proof of the Voronoi Conjecture on Parallelotopes in a New Special Case, Ph.D. Thesis, Queen's University, Ontario, 2005 | MR

[8] B. Delaunay, “Sur la partition régulière de l'espace à 4 dimensions. Première partie”, Izvestiya Akademii nauk SSSR. VII seriya. Otdelenie fiziko-matematicheskikh nauk, 1929, no. 1, 79–110 | Zbl

[9] M. I. Shtogrin, “Pravilnye razbieniya Dirikhle–Voronogo dlya vtoroi triklinnoi gruppy”, Tr. MIAN SSSR, 123, 1973, 3–128 | MR | Zbl

[10] S. S. Ryshkov, E. P. Baranovskii, “$C$-tipy $n$-mernykh reshetok i pyatimernye primitivnye paralleloedry (s prilozheniem k teorii pokrytii)”, Tr. MIAN SSSR, 137, 1976, 3–131 | MR | Zbl

[11] P. Engel, “The contraction types of parallelohedra in $\mathbb E^5$”, Acta Cryst. Sect. A, 56:5 (2000), 491–496 | DOI | MR | Zbl

[12] G. C. Shephard, “Space-filling zonotopes”, Mathematika, 21 (1974), 261–269 | DOI | MR | Zbl

[13] P. McMullen, “Space tiling zonotopes”, Mathematika, 22:2 (1975), 202–211 | DOI | MR | Zbl

[14] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math., 152, Springer-Verlag, New York, 1995 | MR | Zbl

[15] A. P. Poyarkov, A. I. Garber, “O perestanovochnykh mnogogrannikakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2006, no. 2, 3–8 | MR | Zbl

[16] E. S. Fedorov, Nachala ucheniya o figurakh, Tip. Imperatorskoi AN, Sankt-Peterburg, 1885

[17] H. S. M. Coxeter, Regular Polytopes, Dover Publ., New York, 1973 | MR | Zbl

[18] A. N. Magazinov, Lichnaya beseda, 2010

[19] B. A. Venkov, “O proektirovanii paralleloedrov”, Matem. sb., 49(91):2 (1959), 207–224 | Zbl