An Upper Bound for a Valence of a Codimension k Face in a Parallelotope tiling

Details An Upper Bound for a Valence of a Codimension k Face in a Parallelotope tiling An Upper Bound for a Valence of a Codimension k Face in a Parallelotope tiling

2012-01-07

arxiv.org/abs/1201.1539v1

Mathematics

Metric Geometry

8 pages

Scientific paper

Consider a face-to-face parallelotope tiling of $\mathbb R^d$ and a codimension $k$ face $F^{d-k}$ of the tiling. The main result of the present paper is that the valence of $F^{d-k}$ is not greater than $2^k$. If the parallelotope tiles are affinely equivalent to a DV-domain for some lattice (the Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay $k$-cell. However, such an affine equivalence is not assumed in the present proof.

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