CS - 1993 - 09 On Stabbing Lines for Convex Polyhedra in 3

Given a set B of convex polyhedra in R 3 , a linè in R 3 is called a stabbing line for B if it intersects all polyhedra of B. This paper presents an upper bound of O(n 3 logn) on the complexity of the space of stabbing lines for B. We solve a more general problem which counts the number of faces in a set of convex polyhedra, which are implicitly deened by a set of half-spaces and an arrangement of hyperplanes. We show that the former problem is a special case of the latter problem. We also apply this technique to obtain an upper bound on the number of distinct faces that ever appear on the intersection of a set of half-spaces as we insert or delete half-spaces dynamically.