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Belt distance between facets of space-filling zonotopes. (English. Russian original) Zbl 1275.52012
Math. Notes 92, No. 3, 345-355 (2012); translation from Mat. Zametki 92, No. 3, 381-394 (2012).
A belt of a polytope $$P$$ is the set of all facets parallel to a given $$(n-2)$$-face of $$P$$. A sequence of facets is called a belt path if every two consecutive facets in the sequence belong to the same belt. The number of different belt the facets in a belt path belong to is called the length of the path. Finally the belt distance of two facets is the length of the shortest belt path between them and accordingly, the belt diameter of a polytope $$P$$ is the maximal belt distance between any two facets of $$P$$.
The author investigates belt diameters of zonotopes that are also parallelotopes and proves an upper bound of $$\log_2(\frac45 d)$$ for $$d$$-dimensional space-filling zonotopes. To this end, he shows that it is enough to consider zonotopes whose generators lie in two conjugate sets and which behave nicely under projections in a certain direction. Afterwards he uses an inductive argument to show the statement for those special zonotopes.
The author further shows that this bound is sharp in dimensions up to $$6$$.
##### MSC:
 52B11 $$n$$-dimensional polytopes 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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##### References:
 [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, 67–178 (1909). [2] H. Minkowski, ”Allgemeine Lehrsätze über die convexen Polyeder,”Gött. Nachr., 198–219 (1897). · JFM 28.0427.01 [3] P. McMullen, ”Convex bodies which tile space by translation,” Mathematika 27(1), 113–121 (1980). · Zbl 0432.52016 [4] B. A. Venkov, ”On a class of Euclidean polyhedra,” Vestnik Leningrad.Univ. Ser.Mat. Fiz.Him. 9(2), 11–31 (1954). [5] O. K. Zhitomirskii, ”Verschärfung eines Satzes von Voronoi,” Zh. Leningrad. Matem. Obshch. 2, 131–151 (1929). [6] R. M. Erdahl, ”Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra,” European J. Combin. 20(6), 527–549 (1999). · Zbl 0938.52016 [7] A. Ordine, Proof of the Voronoi Conjecture on Parallelotopes in a New Special Case, Ph. D. Thesis (Queen’s University, Ontario, 2005). [8] B. Delaunay, ”Sur la partition régulière de l’espace à 4 dimensions.Deuxième partie,” Izv. Akad. Nauk SSSR Ser. VII. Otd. Fiz.Mat. Nauk, No. 2, 147–164 (1929). [9] M. I. Shtogrin, ”Regular Dirichlet-Voronoi partitions for the second triclinic group,” Trudy Mat. Inst. Steklov 123, 3–128 (1973) [Proc. Steklov Inst. Math. 123, 1–116 (1973)]. [10] S. S. Ryshkov and E. P. Baranovskii, ”S-types of n-dimensional lattices and five-dimensional primitive parallelohedra (with an application to covering theory),” Trudy Mat. Inst. Steklov 137, 3–131 (1976) [Proc. Steklov Inst.Math. 137, 1–140 (1976)]. · Zbl 0419.10031 [11] P. Engel, ”The contraction types of parallelohedra in $$\mathbb{E}$$ 5,” Acta Cryst. Sec. A 56(5), 491–496 (2000). · Zbl 1188.52021 [12] G. C. Shephard, ”Space-filling zonotopes,” Mathematika 21, 261–269 (1974). · Zbl 0296.52004 [13] P. McMullen, ”Space tiling zonotopes,” Mathematika 22(2), 202–211 (1975). · Zbl 0316.52005 [14] G.M. Ziegler, Lectures on Polytopes, in Grad. Texts inMath. (Springer-Verlag, NewYork, 1995), Vol. 152. [15] A. P. Poyarkov and A. I. Garber, ”On permutohedra,” Vestnik Moskov.Univ. Ser. I Mat. Mekh., No. 2, 3–8 (2006) [Moscow Univ.Math. Bull. 61 (2), 1–6 (2006)]. [16] E. S. Fedorov, Foundations of the Theory of Shapes (Emperor’s Academy of Sciences Press, St.-Petersburg, 1885) [in Russian]. [17] H. S.M. Coxeter, Regular Polytopes (Dover Publ., New York, 1973). [18] A. N. Magazinov, personal communication (2010). [19] B. A. Venkov, ”On the projections of parallelohedra,” Mat. Sb. 49(91)(2), 207–224 (1959).
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