# Counting facets and incidences

- First Online:

- Received:
- Revised:

DOI: 10.1007/BF02187848

- Cite this article as:
- Agarwal, P.K. & Aronov, B. Discrete Comput Geom (1992) 7: 359. doi:10.1007/BF02187848

## Abstract

We show that*m* distinct cells in an arrangement of*n* planes in ℝ^{3} are bounded by*O*(*m*^{2/3}*n*+*n*^{2}) faces, which in turn yields a tight bound on the maximum number of facets bounding*m* cells in an arrangement of*n* hyperplanes in ℝ^{d}, for every*d*≥3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ℝ^{3}. We also present a simpler proof of the*O*(*m*^{2/3}*n*^{d/3}+*n*^{d−1}) bound on the number of incidences between*n* hyperplanes in ℝ^{d} and*m* vertices of their arrangement.