Voronoi's Conjecture for extensions of Voronoi parallelohedra

Abstract

Let $I$ be a segment in the $d$-dimensional Euclidean space ${\mathbb{E}}^d$. Let $P$ and $P+I$ be parallelohedra in ${\mathbb{E}}^d$, where "+" denotes the Minkowski sum. We prove that Voronoi's Conjecture holds for $P+I$, i.e. $P+I$ is a Voronoi parallelohedron for some Euclidean metric in ${\mathbb{E}}^d$, if Voronoi's Conjecture holds for $P$.

27 pages, 4 figures

References in corpus (1)

• On the sum of the Voronoi polytope of a lattice with a zonotope

Cited by in corpus (2)

• Voronoi conjecture for five-dimensional parallelohedra
• Proof of the Voronoi conjecture for 3-irreducible parallelotopes