Minkowski sum of a Voronoi parallelotope and a segment

TL;DR

This paper characterizes when the Minkowski sum of a Voronoi parallelotope and a segment results in another Voronoi parallelotope, linking the segment's direction to the dual set of facet normals.

Contribution

It provides a necessary and sufficient condition for the Minkowski sum of a Voronoi parallelotope and a segment to be a Voronoi parallelotope, involving the dual set of facet normals.

Findings

Minkowski sum of a Voronoi parallelotope and a segment is a Voronoi parallelotope under specific conditions.

The segment must be parallel to a vector in the dual of the set of facet normals.

The resulting Voronoi parallelotope has a quadratic form modified by a rank-1 form related to the 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.