An upper bound for a valence of a face in a parallelohedral tiling |
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Authors: | Alexander Magazinov |
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Institution: | Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkina street, Moscow 119991, Russia; Yaroslavl State University, 14 Sovetskaya street, Yaroslavl 150000, Russia |
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Abstract: | Consider a face-to-face parallelohedral tiling of Rd 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 2k. 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. |
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