Voronoi's Conjecture for Extensions of Voronoi Parallelohedra
In 1908 Voronoi conjectured that every parallelohedron is a Voronoi parallelohedron for some Euclidean metric in E^d . Although the conjecture is still neither proved, nor disproved, there are several positive results for some special classes of parallelohedra. In this paper we extend the list of such classes by one new case. Let I be a segment in the d-dimensional Euclidean space E^d . Let P and P + I be parallelohedra in E^d , where the plus sign denotes the Minkowski sum. We prove that, if Voronoi’s Conjecture holds for P , then Voronoi’s Conjecture holds for P + I as well.