ACM Computing Surveys (CSUR)
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\def\B{{\cal B}} \def\reals{{\bf R}} Given a set $\B$ of convex polyhedra in $\reals^3$, a line $\ell$ in $\reals^3$ is called a {\em stabbing line\/} for $\B$ if it intersects all polyhedra of $\B$. %Pellegrini and Shor \cite{PeS} proved that the %combinatorial complexity of the space of stabbing lines %for $\B$ is $O(n^3 2^{\sqrt{\log n}})$, where $n$ is the total number of %edges in the polyhedra of $\B$. This paper presents an upper bound of $O(n^3 \log n)$ 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 defined 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.