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Title: 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:
 [1]
  1. 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}
}