Tessellations generated by hyperplanes
Let \({\mathcal H}\) be a locally finite system of hyperplanes in \({\mathbb{R}}^ d\) such that the cells of the induced cell complex decomposition \({\mathcal T}\) of \({\mathbb{R}}^ d\) have uniformly bounded diameters. If \({\mathcal H}\) is simple and the density of the vertices in \({\mathcal T}\) exists, then the density of the k-cells in \({\mathcal T}\) exists and further the mean number of j-faces of the k-cells in \({\mathcal T}\) exists. The author shows that these numbers are exactly those occurring from the cubical tesselation, generated by hyperplanes parallel to the coordinate hyperplanes. For certain nonsimple systems \({\mathcal H}\) corresponding inequalities are obtained.
scientific article; zbMATH DE number 3992331