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An upper bound for a valence of a face in a parallelohedral tiling
Consider a face-to-face parallelohedral tiling of and a
-dimensional face of the tiling. We prove that the valence of
(i.e. the number of tiles containing as a face) is not greater than .
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 -cell. Yet we emphasize that such an affine
equivalence is not assumed in the proof.Comment: 10 page