An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R d and a ( d − k ) -dimensional face F of the tiling. We prove that the valence of F (i.e. the number of tiles containing F as a face) is not greater than 2 k . If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k -cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.