Abstract
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 \(\varepsilon > 0\).
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Ezra, E., Sharir, M. A Single Cell in an Arrangement of Convex Polyhedra in \({\Bbb R}^3\). Discrete Comput Geom 37, 21–41 (2007). https://doi.org/10.1007/s00454-006-1272-9
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DOI: https://doi.org/10.1007/s00454-006-1272-9