Minkowski Sum of a Voronoi Parallelotope and a Segment

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 wae(p)=w as right hand side and P(a) + z(u) = P(a + wae). A similar assertion is true for Minkowski sum of a reducible Voronoi parallelotope with a segment.