On Cell Complexities in Hyperplane Arrangements

Let H be a collection of n hyperplanes in d-space. We will assume that the planes are in general position, meaning that any k planes meet in a d− k-flat, if k = 1, . . . , d, and not at all if k > d. It is not difficult to see that worst-case cell complexity can always be achieved by planes in general position. Let P be a set of m points, not lying on any hyperplane. Denote by K (d) j (P,H) the number of j-faces bounding the cells of A(H) that contain points of P . We will mainly be concerned with the case j = ⌈d/2⌉, because, as follows from the Dehn-Sommerville relations (see, e.g., [3]), the total number of faces, of all dimensions, of a cell (which is a simple d-polytope) is at most proportional to the number of its ⌈d/2⌉-faces. We denote by K (d) j (m,n) the maximum of K (d) j (P,H) over all sets P,H as above. Work on this paper has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Boris Aronov was also supported by NSF Grant CCR–99-72568. Work by Micha Sharir was also supported by NSF Grant CCR–97-32101, by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), by the ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Part of the work by Boris Aronov on the paper was done when he visited Tel Aviv University in May 2000. Department of Computer and Information Science, Polytechnic University, Brooklyn, NY 11201-3840, USA. E-mail: [email protected] School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. E-mail: [email protected]