Geodesic
The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice $E_7$
Sbornik. Mathematics, Tome 203 (2012) no. 11, pp. 1571-1588 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that the Minkowski sum $P_{\mathrm V}(E_7)+Z(U)$ of the Voronoi polytope $P_{\mathrm V}(E_7)$ of the root lattice $E_7$ and the zonotope $Z(U)$ is a 7-dimensional parallelohedron if and only if the set $U$ consists of minimal vectors of the dual lattice $E_7^*$ up to scalar multiplication, and $U$ does not contain forbidden sets. The minimal vectors of $E_7$ are the vectors $r$ of the classical root system $\mathbf E_7$. If the $r^2$-norm of the roots is set equal to 2, then the scalar products of minimal vectors from the dual lattice only take the values $\pm1/2$. A set of minimal vectors is referred to as forbidden if it consists of six vectors, and the directions of some of these vectors can be changed so as to obtain a set of six vectors with all the pairwise scalar products equal to $1/2$. Bibliography: 11 titles.
Keywords: Minkowski sum, Voronoi polytope, zonotope, unimodular set, matroid. 
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V. P. Grishukhin. The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice $E_7$. Sbornik. Mathematics, Tome 203 (2012) no. 11, pp. 1571-1588. http://geodesic.mathdoc.fr/item/SM_2012_203_11_a2/

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

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

[3] V. P. Grishukhin, “Koe-chto o reshetkakh $E_6$ i $E_6^*$”, Chebyshevskii sb., 7:2 (2006), 78–94 | MR | Zbl

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

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

[6] A. Björner, M. Las Vergnas, B. Stumfels, N. White, G. M. Ziegler, Oriented matroids, Encyclopedia Math. Appl., 46, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[7] M. Deza, V. Grishukhin, “Voronoi's conjecture and space tiling zonotopes”, Mathematika, 51:1–2 (2004), 1–10 | DOI | MR | Zbl

[8] P. D. Seymour, “Decomposition of regular matroids”, J. Combin. Theory Ser. B, 28:3 (1980), 305–359 | DOI | MR | Zbl

[9] M. Aigner, Combinatorial theory, Grundlehren Math. Wiss., 234, Springer-Verlag, Berlin–New York, 1979 | MR | MR | Zbl

[10] V. Danilov, V. Grishukhin, “Maximal unimodular systems of vectors”, European J. Combin., 20:6 (1999), 507–526 | DOI | MR | Zbl

[11] V. P. Grishukhin, V. I. Danilov, G. A. Koshevoi, “Unimodular systems of vectors are embeddable in the $(0,1)$-cube”, Math. Notes, 88:6 (2010), 891–893 | DOI | MR | Zbl