Minkowski sum of a Voronoi parallelotope and a segment

Abstract

By a {\em Voronoi parallelotope} $P(a)$ we mean a parallelotope determined by a non-negative quadratic form $a$. It was studied by Voronoi in his famous memoir. For a set of vectors $\mathcal P$, we call its {\em dual} a set of vectors ${\mathcal P}^*$ such that $\langle p,q\rangle\in\{0,\pm 1\}$ for all $p\in{\mathcal P}$ and $q\in{\mathcal P}^*$. We prove that Minkowski sum of a Voronoi parallelotope $P(a)$ and a segment is a Voronoi parallelotope $P(a+a_e)$ if and only if this segment is parallel to a vector $e$ of the dual of the set of normal vectors of all facets of $P(a)$, where $a_e(p)=b\langle e,p\rangle^2$ is a quadratic form of rank 1 related to the segment.