# The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice E{sub 7}

## Abstract

We show that the Minkowski sum P{sub V}(E{sub 7})+Z(U) of the Voronoi polytope P{sub V}(E{sub 7}) of the root lattice E{sub 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{sub 7}{sup *} up to scalar multiplication, and U does not contain forbidden sets. The minimal vectors of E{sub 7} are the vectors r of the classical root system E{sub 7}. If the r{sup 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 {+-}1/2. 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.

- Authors:

- Central Economics and Mathematics Institute, RAS, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22156587

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 203; Journal Issue: 11; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; MANY-DIMENSIONAL CALCULATIONS; MINKOWSKI SPACE; SCALARS; VECTORS

### Citation Formats

```
Grishukhin, Vyacheslav P.
```*The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice E{sub 7}*. United States: N. p., 2012.
Web. doi:10.1070/SM2012V203N11ABEH004276.

```
Grishukhin, Vyacheslav P.
```*The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice E{sub 7}*. United States. doi:10.1070/SM2012V203N11ABEH004276.

```
Grishukhin, Vyacheslav P. Fri .
"The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice E{sub 7}". United States. doi:10.1070/SM2012V203N11ABEH004276.
```

```
@article{osti_22156587,
```

title = {The Minkowski sum of a zonotope and the Voronoi polytope of the root lattice E{sub 7}},

author = {Grishukhin, Vyacheslav P},

abstractNote = {We show that the Minkowski sum P{sub V}(E{sub 7})+Z(U) of the Voronoi polytope P{sub V}(E{sub 7}) of the root lattice E{sub 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{sub 7}{sup *} up to scalar multiplication, and U does not contain forbidden sets. The minimal vectors of E{sub 7} are the vectors r of the classical root system E{sub 7}. If the r{sup 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 {+-}1/2. 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.},

doi = {10.1070/SM2012V203N11ABEH004276},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 11,

volume = 203,

place = {United States},

year = {2012},

month = {11}

}