Tessellations generated by hyperplanes

Abstract

Let ℋ be a locally finite system of hyperplanes in ℝd with the property that the cells of the induced cell complex decomposition ℐ of ℝd have uniformly bounded diameters. If ℋ is simple and the density of the vertices in ℐ exists, then the density of the k-cells in ℐ exists and can be given explicitly (k = 1, ..., d). Also, the mean number of j-faces of the k-cells in ℐ exists and can be calculated. For certain nonsimple systems ℋ, corresponding inequalities are obtained. © 1987 Springer-Verlag New York Inc.