A Single Cell in an Arrangement of Convex Polyhedra in ${\Bbb R}^3$

• Authors: Esther Ezra;Micha Sharir
• Venue: Discrete & Computational Geometry
• Year: 2007

We show that the combinatorial complexity of a single cell in an arrangement of k convex polyhedra in 3-space having n facets in total is $O(nk^{1+\varepsilon})$, for any $\varepsilon 0$, thus settling a conjecture of Aronov et al. We then extend our analysis and show that the overall complexity of the zone of a low-degree algebraic surface, or of the boundary of an arbitrary convex set, in an arrangement of k convex polyhedra in 3-space with n facets in total, is also $O(nk^{1+\varepsilon})$, for any $\varepsilon 0$. Finally, we present a deterministic algorithm that constructs a single cell in an arrangement of this kind, in time $O(nk^{1+\varepsilon} \log^3{n})$, for any $\varepsilon 0$.