Geodesic
Lifting of parallelohedra
Sbornik. Mathematics, Tome 210 (2019) no. 10, pp. 1434-1455 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A parallelohedron is a polyhedron that can tessellate the space via translations without gaps and overlaps. Voronoi conjectured that any parallelohedron is affinely equivalent to a Dirichlet-Voronoi cell of some lattice. Delaunay used the term displacement parallelohedron in his paper “Sur la tiling régulière de l'espace à 4 dimensions. Première partie”, where the four-dimensional parallelohedra are listed. In our work, such a parallelohedron is called a lifted parallelohedron, since it is obtained as an extension of a parallelohedron to a parallelohedron of dimension larger by one. It is shown that the operation of lifting yields precisely parallelohedra whose Minkowski sum with some nontrivial segment is again a parallelohedron. It is proved that Voronoi's conjecture holds for parallelohedra admitting lifts and lifted in general position. Bibliography: 20 titles.
Keywords: parallelohedral tiling, lattice, free direction, generatrissa, lamina. 
@article{SM_2019_210_10_a4,
     author = {V. P. Grishukhin and V. I. Danilov},
     title = {Lifting of parallelohedra},
     journal = {Sbornik. Mathematics},
     pages = {1434--1455},
     year = {2019},
     volume = {210},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_10_a4/}
}
TY  - JOUR
AU  - V. P. Grishukhin
AU  - V. I. Danilov
TI  - Lifting of parallelohedra
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 1434
EP  - 1455
VL  - 210
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_10_a4/
LA  - en
ID  - SM_2019_210_10_a4
ER  - 
%0 Journal Article
%A V. P. Grishukhin
%A V. I. Danilov
%T Lifting of parallelohedra
%J Sbornik. Mathematics
%D 2019
%P 1434-1455
%V 210
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2019_210_10_a4/
%G en
%F SM_2019_210_10_a4
V. P. Grishukhin; V. I. Danilov. Lifting of parallelohedra. Sbornik. Mathematics, Tome 210 (2019) no. 10, pp. 1434-1455. http://geodesic.mathdoc.fr/item/SM_2019_210_10_a4/

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

[2] V. P. Grishukhin, “Parallelotopes of non-zero width”, Sb. Math., 195:5 (2004), 669–686 | DOI | DOI | MR | Zbl

[3] Á. G. Horváth, “On the connection between the projection and the extension of a parallelotope”, Monatsh. Math., 150:3 (2007), 211–216 | DOI | MR | Zbl

[4] A. Magazinov, “Voronoi's conjecture for extensions of Voronoi parallelohedra”, Mosc. J. Comb. Number Theory, 5:3 (2015), 86–131 | MR | Zbl

[5] B. Delaunay, “Sur la partition régulière de l'espace à 4 dimensions. Première partie”, Izv. AN SSSR. VII ser. Otd. fiz.-matem. nauk, 1929, no. 1, 79–110 ; Deuxième partie, no. 2, 147–164 | Zbl

[6] G. Voronoi, “Nouvelles applications des paramètres continus à là théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs”, J. Reine Angew. Math., 1908:134 (1908), 198–287 ; 1909:136 (1909), 67–178 | DOI | MR | Zbl | DOI | MR | Zbl

[7] C. Davis, “The set of non-linearity of a convex piecewise-linear function”, Scripta Math., 24 (1959), 219–228 | MR | Zbl

[8] P. McMullen, “Duality, sections and projections of certain Euclidean tilings”, Geom. Dedicata, 49:2 (1994), 183–202 | DOI | MR | Zbl

[9] J. H. Conway, N. J. A. Sloane, “The cell structures of certain lattices”, Miscellanea mathematica, Springer, Berlin, 1991, 71–107 | MR | Zbl

[10] S. S. Ryshkov, “A direct geometric description of the $n$-dimensional Voronoi parallelohedra of second type”, Russian Math. Surveys, 54:1 (1999), 264–265 | DOI | DOI | MR | Zbl

[11] N. P. Dolbilin, “Properties of faces of parallelohedra”, Proc. Steklov Inst. Math., 266:1 (2009), 105–119 | DOI | MR | Zbl

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

[13] H. Seifert, W. Threlfall, Lehrbuch der Topologie, B. G. Teubner, Leipzig, 1934, iv+353 pp. | Zbl

[14] J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren Math. Wiss., 290, Springer-Verlag, New York, 1988, xxviii+663 pp. | DOI | MR | MR | MR | Zbl

[15] E. P. Baranovskii, “Partitioning of Euclidean spaces into $L$-polytopes of some perfect lattices”, Proc. Steklov Inst. Math., 196 (1992), 29–51 | MR | Zbl

[16] S. S. Ryshkov, E. P. Baranovskii, “$C$-types of $n$-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)”, Proc. Steklov Inst. Math., 137 (1976), 1–140 | MR | Zbl

[17] A. Végh, “On extraction of parallelotopes”, Stud. Univ. Žilina. Math. Ser., 26:1 (2014), 49–54 | MR | Zbl

[18] M. Dutour Sikirić, V. Grishukhin, A. Magazinov, “On the sum of a parallelotope and a zonotope”, European J. Combin., 42 (2014), 49–73 | DOI | MR | Zbl

[19] V. P. Grishukhin, “Free and nonfree Voronoi polyhedra”, Math. Notes, 80:3 (2006), 355–365 | DOI | DOI | MR | Zbl

[20] V. P. Grishukhin, “Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice $E_7^*$”, Proc. Steklov Inst. Math., 275 (2011), 60–77 | DOI | MR | Zbl