Geodesic
Parallelotopes of non-zero width
Sbornik. Mathematics, Tome 195 (2004) no. 5, pp. 669-686 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In 1959, Venkov introduced a concept of polytope of non-zero width in the direction of a subspace and studied parallelotopes of non-zero width. In the present paper properties of a parallelotope of non-zero width in the direction of a straight line are investigated. In particular, it is proved that a parallelotope of non-zero width in the direction of a straight line is the Minkowski sum of a parallelotope of width zero and a segment of this line. Necessary and sufficient conditions ensuring that the sum of a parallelotope and a line segment is again a parallelotope are presented. 
@article{SM_2004_195_5_a2,
     author = {V. P. Grishukhin},
     title = {Parallelotopes of non-zero width},
     journal = {Sbornik. Mathematics},
     pages = {669--686},
     year = {2004},
     volume = {195},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2004_195_5_a2/}
}
TY  - JOUR
AU  - V. P. Grishukhin
TI  - Parallelotopes of non-zero width
JO  - Sbornik. Mathematics
PY  - 2004
SP  - 669
EP  - 686
VL  - 195
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2004_195_5_a2/
LA  - en
ID  - SM_2004_195_5_a2
ER  - 
%0 Journal Article
%A V. P. Grishukhin
%T Parallelotopes of non-zero width
%J Sbornik. Mathematics
%D 2004
%P 669-686
%V 195
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2004_195_5_a2/
%G en
%F SM_2004_195_5_a2
V. P. Grishukhin. Parallelotopes of non-zero width. Sbornik. Mathematics, Tome 195 (2004) no. 5, pp. 669-686. http://geodesic.mathdoc.fr/item/SM_2004_195_5_a2/

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

[2] Ryshkov S. S., Baranovskii E. P., $C$-tipy $n$-mernykh reshetok i pyatimernye primitivnye paralleloedry (s prilozheniem k teorii pokrytii), Trudy MIAN, 137, 1976 | MR | Zbl

[3] Deza M., Grishukhin V., “Rank 1 forms, closed zones and laminae”, J. Théor. Nombres Bordeaux, 14 (2002), 103–112 | MR | Zbl

[4] Engel P., “Investigations of parallelohedra in $\mathbb R^d$”, Voronoi's impact on modern science, 2, eds. P. Engel, H. Syta, Institute of Mathematics, Kyiv, 1998, 22–60 | Zbl

[5] Delaunay B. N., “Sur la partition régulière de l'espace á 4 dimensions”, Izv. AN SSSR. Ser. matem., 1929, no. 1, 79–110 | Zbl

[6] Aigner M., Kombinatornaya teoriya, Mir, M., 1982 | MR

[7] Venkov B. A., “Ob odnom klasse evklidovykh mnogogrannikov”, Vestn. LGU, 1954, no. 2, 11–31 | MR

[8] Conway J. H., Sloane N. J. A., Sphere packings, lattices and groups, Grundlehren Math. Wiss., 290, Springer-Verlag, Berlin, 1988 | MR | Zbl

[9] Baranovskii E. P., Grishukhin V. P., “Non-rigidity degree of a lattice and rigid lattices”, European J. Combin., 22 (2001), 921–935 | DOI | MR | Zbl

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

[11] Erdahl E., Ryshkov S. S., “On lattice dicing”, European J. Combin., 15 (1994), 459–481 | DOI | MR | Zbl

[12] Danilov V. I., Grishukhin V. P., “Maximal unimodular systems of vectors”, European J. Combin., 20 (1999), 507–526 | DOI | MR | Zbl