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Permutable polyhedra. (Russian, English) Zbl 1130.52007
Vestn. Mosk. Univ., Ser. I 2006, No. 2, 3-8 (2006); translation in Mosc. Univ. Math. Bull. 61, No. 2, 1-6 (2006).
The author considers the known properties of a permutation polyhedron and in terms of these properties prove that it is a parallelohedron. Moreover, it is the Dirichlet-Voronoi polyhedron for the first type main lattice. This provides the other approach to the description of some properties of a parallelohedron being a permutation polyhedron, obtained by S. S. Ryshkov [Sov. Math., Dokl. 3, 1451–1454 (1962); translation from Dokl. Akad. Nauk SSSR 146, 1027–1030 (1962; Zbl 0152.20603)]. In the paper it is also proved that among all permutation polyhedra only those similar to the main permutation polyhedron are parallelohedra.

##### MSC:
 52B11 $$n$$-dimensional polytopes 52B70 Polyhedral manifolds
##### Keywords:
Dirichlet-Voronoi polyhedron