  Consider a face-to-face parallelohedral tiling of $\mathbb R^d$ 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.
