## Abstract

We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of k convex polyhedra in 3-space having n facets in total. We use a variant of the technique of Halperin and Sharir [17], and show that this complexity is O(nk^{1+ε}), for any ε > 0, thus almost 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(nk^{1+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(nk^{1+ε} log^{2} n), for any ε > 0.

Original language | English |
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Pages | 22-31 |

Number of pages | 10 |

DOIs | |

State | Published - 2005 |

Externally published | Yes |

Event | 21st Annual Symposium on Computational Geometry, SCG'05 - Pisa, Italy Duration: 6 Jun 2005 → 8 Jun 2005 |

### Conference

Conference | 21st Annual Symposium on Computational Geometry, SCG'05 |
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Country/Territory | Italy |

City | Pisa |

Period | 6/06/05 → 8/06/05 |

## Keywords

- Algorithms
- Theory

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