An extension of Voronïʹs theorem on primitive parallelotopes

Voronoï proved that any parallelotope that is the prototile of a face-to-face primitive tiling is the affine image of a Voronoï cell. We prove that the affinity is unique (up to an orthogonal transformation). Using a theorem of Zhitomirskii, this result can be extended to parallelotopes with maximal numbers of faces.