Geodesic
Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice~$E_7^*$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 68-86.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi polytopes of the root lattice $E_7$ and the dual lattice $E_7^*$. Here three uniform polytopes related to the Coxeter–Dynkin diagram of the Lie algebra $E_7$ play a special role. These are the Gosset polytope $P_\mathrm{Gos}=3_{21}$, which is a Delaunay polytope, the contact polytope $2_{31}$ (both for the lattice $E_7$), and the Voronoi polytope $P_\mathrm V(E_7^*)=1_{32}$ of the dual lattice $E_7^*$. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi polytopes. 
@article{TM_2011_275_a3,
     author = {V. P. Grishukhin},
     title = {Delaunay and {Voronoi} polytopes of the root lattice $E_7$ and of the dual lattice~$E_7^*$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {68--86},
     publisher = {mathdoc},
     volume = {275},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_275_a3/}
}
TY  - JOUR
AU  - V. P. Grishukhin
TI  - Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice~$E_7^*$
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2011
SP  - 68
EP  - 86
VL  - 275
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2011_275_a3/
LA  - ru
ID  - TM_2011_275_a3
ER  - 
%0 Journal Article
%A V. P. Grishukhin
%T Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice~$E_7^*$
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2011
%P 68-86
%V 275
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2011_275_a3/
%G ru
%F TM_2011_275_a3
V. P. Grishukhin. Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice~$E_7^*$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 68-86. http://geodesic.mathdoc.fr/item/TM_2011_275_a3/

[1] Baranovskii E.P., “Razbienie evklidovykh prostranstv na L-mnogogranniki nekotorykh sovershennykh reshetok”, Tr. MIAN, 196, 1991, 27–46 | MR

[2] Baranovskii E.P., “The perfect lattices $\Gamma (\mathfrak A^n)$, and the covering density of $\Gamma (\mathfrak A^9)$”, Eur. J. Comb., 15 (1994), 317–323 | DOI | MR | Zbl

[3] Brouwer A.E., Cohen A.M., Neumaier A., Distance-regular graphs, Springer, Berlin, 1989 | MR | Zbl

[4] Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups, Grundl. math. Wissensch., 290, Springer, Berlin, 1988 | DOI | MR | Zbl

[5] Conway J.H., Sloane N.J.A., “Low-dimensional lattices. II: Subgroups of $\mathrm {GL}(n,\mathbb Z)$”, Proc. Roy. Soc. London A, 419 (1988), 29–68 | DOI | MR | Zbl

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

[7] Coxeter H.S.M., “The pure Archimedian polytopes in six and seven dimensions”, Math. Proc. Cambridge Philos. Soc., 24 (1928), 1–9 | DOI | Zbl

[8] Coxeter H.S.M., “Extreme forms”, Can. J. Math., 3 (1951), 391–441 | DOI | MR | Zbl

[9] Coxeter H.S.M., Regular polytopes, 3rd ed., Dover, New York, 1973 | MR

[10] Coxeter H.S.M., “The evolution of Coxeter–Dynkin diagrams”, Nieuw Arch. Wiskd. Ser. 4., 9 (1991), 233–248 | MR | Zbl

[11] Deza M.M., Laurent M., Geometry of cuts and metrics, Springer, Berlin, 1997 | MR | Zbl

[12] Dolbilin N.P., “Svoistva granei paralleloedrov”, Tr. MIAN, 266, 2009, 112–126 | MR | Zbl

[13] Gosset T., “On the regular and semi-regular figures in space of $n$ dimensions”, Messenger Math., 29 (1900), 43–48

[14] Grishukhin V.P., “Paralleloedry nenulevoi tolschiny”, Mat. sb., 195:5 (2004), 59–78 | DOI | MR | Zbl

[15] Pervin E., “The Voronoi cells of the $E_6^*$ and $E_7^*$ lattices”, Coding theory, design theory, group theory, Proc. Marshall Hall Conf., Vermont, Sept. 13–18, 1990, J. Wiley Sons, New York, 1993, 243–248 http://home.digital.net/~pervin/publications/vermont.html | MR

[16] Worley R.T., “The Voronoi region of $E_7^*$”, SIAM J. Discr. Math., 1 (1988), 134–141 | DOI | MR | Zbl