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 $P$.