A Single Cell in an Arrangement of Convex Polyhedra in R

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(nk1+ε), for any ε > 0, thus settling a conjecture of Aronov et al. [5]. 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(nk1+ε), for any ε > 0. Finally, we present a deterministic algorithm that constructs a single cell in an arrangement of this kind, in time O(nk1+ε log n), for any ε > 0.