### Journals Information

**
Mathematics and Statistics Vol. 3(6), pp. 151 - 156 DOI: 10.13189/ms.2015.030603 Reprint (PDF) (91Kb) **

## Minkowski Sum of a Voronoi Parallelotope and a Segment

**Robert Erdahl ^{1}, Viacheslav Grishukhin ^{2}^{,*}**

^{1}Queen's University, Kingston, Canada

^{2}CEMI Russian Academy of Sciences, Nakhimovskii prosp.47 117418 Moscow, Russia

**ABSTRACT**

By a Voronoi parallelotope P(a) we mean a parallelotope determined by linear in normal vectors p inequalities with a non-negative quadratic form a(p) as right hand side. For a positive form a, it was studied by Voronoi in his famous memoir. For a set of vectors P, we call its dual a set of vectors P^{*} such that ∈ {0;±1} for all p ∈ P and q ∈ P^{*}. We prove that Minkowski sum of an irreducible Voronoi parallelotope P(a) and a segment z(u) is a Voronoi parallelotope if and only if u = we, where w > 0 and e is a vector of the dual of the set of normal vectors of all facets of P(a). Then the segment z(u) is described by the same set of inequalities with wa_{e}(p)=w as right hand side and P(a) + z(u) = P(a + wa_{e}). A similar assertion is true for Minkowski sum of a reducible Voronoi parallelotope with a segment.

**KEYWORDS**

Parallelotope, Voronoi Parallelotope, Minkowski Sum, Dual Set

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Robert Erdahl , Viacheslav Grishukhin , "Minkowski Sum of a Voronoi Parallelotope and a Segment," Mathematics and Statistics, Vol. 3, No. 6, pp. 151 - 156, 2015. DOI: 10.13189/ms.2015.030603.

(b). APA Format:

Robert Erdahl , Viacheslav Grishukhin (2015). Minkowski Sum of a Voronoi Parallelotope and a Segment. Mathematics and Statistics, 3(6), 151 - 156. DOI: 10.13189/ms.2015.030603.