An analog of Ordin's theorem for parallelotopes

V. P. Grishukhin

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 407-420.

Résumé

Parallelotope is a convex polytope in an affine space such that its shifts by vectors of a lattice $L$ fill the space without gaps and intersections by inner points. A special case of a parallelotope is a Dirichlet-Voronoi cell of a lattice with respect to a metric generated by a positive quadratic form. More than 100 years ago G. Voronoi supposed that each parallelotope is a Dirichlet-Voronoi cell of its lattice with respect some metric. A.Ordin introduced notions of an irreducible face and a $k$-irreducible parallelotope whose all faces of codimension $K$ are irreducible. A parallelotope tiling is called $k$-irreducible if its parallelotopes are $k$-irreducible. Ordin proved the conjecture of Voronoi for $3$-irreducible parallelotopes. There are two vectors related to a facet $F$ of a parallelotope. Namely, facet vector $l_F$ of the lattice $L$ of the tiling $\mathcal T$ and normal vector $p_F$ of the facet $F$. The facet vectors integrally generate the lattice $L$. One of the form of Voronoi conjecture asserts that there are such parameters $s(F)$ that scaled (canonical) normal vectors $s(F)p_F$ integrally generate a lattice $\Lambda$. In this paper, uniquely scaled faces are defined. Such a face $G$ determines uniquely up to a multiple parameters $s(F)$ of facets of the tiling $\mathcal T$ containing the face $G$. A tiling whose faces of codimension $k$ are uniquely scaled is $k$-irreducible. It is proved here the following analog of Ordin's Theorem: There exists a canonical scaling of normal vectors of facets of the tiling $\mathcal T$ if, for some integer $k\ge 1$, all its faces of codimension $k$ and $k+1$ are uniquely scaled. The cases $k=2$ and $k=3$ correspond to $2$- and $3$-irreducible tilings of Ordin.

Détail

Keywords: parallelotope, Voronoi conjecture, uniquely scaled normal vectors.

Bibliographie

[1] B. A. Venkov, “Ob odnim klasse ehvklidovyh mnogogrannikov”, Vestnik LGU, 1954, no. 2, 11–31

[2] N. P. Dolbilin, “Svojstva granej paralleloehdrov”, Trudy MIAN, 266, 2009, 112–126 | Zbl

[3] G. F. Voronoi, “Nouvelles applications des paramètres continus à la théorie de formes quadratiques Deuxième mémoire”, J. Reine. Angew. Math., 134 (1908), 198–287 ; 136 (1909), 67–178 ; Voronoi G.F., “Issledovanie o primitivnykh paralleloedrakh”, Sobr. soch., v. 2, Izd-vo AN USSR, Kiev, 1952, 239–368 | DOI | MR | Zbl | DOI | MR | Zbl

[4] M. Deza, V. Grishukhin, “Properties of parallelotopes equivalent to Voronoi's conjecture”, Europ. J. Combinatorics, 25 (2004), 517–533 | DOI | MR | Zbl

[5] V. P. Grishuhin, “Paralleloehdry, opredelyaemye kvadratichnoj formoj”, Trudy MIAN, 288, 2015, 81–93 | DOI | Zbl

[6] A. Ordin, A proof of Voronoi conjecture on parallelotopes in a new special case, Ph.D. thesis, Queen's University, Kingston, 2005, 141 pp. | MR

[7] Z̆itomirskij O. K. (Zhitomirskii), “Verschärfung eines Satzes von Woronoi”, Zhurnal Leningrdskogo Matematicheskogo Obshchestva, 2 (1929), 131–151

[8] Ryshkov S. S., Rybnikov K. A. (Jr.), “The theory of quality translations with applications to tilings”, Europ. J. of Combinatorics, 18:4 (1997), 431–444 | DOI | MR | Zbl

[9] A. A. Gavrilyuk, “Geometriya podemov razbienij evklidovyh prostranstv”, Trudy MIAN, 288, 2015, 49–66 | DOI | Zbl

[10] A. Garber, A. Gavrilyuk, A. Magazinov, “The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface”, Discrete Compute. Geom., 53:2 (2015), 245–260 | DOI | MR | Zbl

[11] Á. G. Horváth, “On the boundary of an extremal body”, Beiträge zur Algebra und Geometrie, 40:2 (1999), 331–342 | MR | Zbl

[12] B. N. Delaunay, “Sur la partition rǵulière de l'espace á 4 dimensions. I; II”, Izv. AN SSSR, VII ser. Otd. fiz-matem. nauk, 1929, no. 1, 79–110 ; no. 2, 147–164 | Zbl

[13] A. N. Magazinov, “On Delaunay's theorem classifying coincidences of parallelohedra at faces of codimension 3”, Modelling and Analysis of inform. systems, 20:4 (2013), 71–80

[14] G. Zejfert, V. Trell'fal', Topologiya, ONTI,, M.–L., 1938, 448 pp.