### Abstract

We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of k convex polygons in the plane, with a total of n edges. We show: (1) The maximum complexity of the entire common exterior is Θ(na(k) + k^{2}). ^{2} (2) The maximum complexity of a single cell of the common exterior is Θ(na(k)). (3) The complexity of m distinct cells in the common exterior is O(m^{2/3}k^{2/3}log^{1/3}(k^{2}/m) + n log k) and can be Ω(m^{2/3}k^{2/3} + na(k)) in the worst case.

Original language | English (US) |
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Pages (from-to) | 139-149 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 8 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1997 |

### Keywords

- Arrangements
- Combinatorial complexity
- Common exterior
- Convex polygons

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

Aronov, B., & Sharir, M. (1997). The common exterior of convex polygons in the plane.

*Computational Geometry: Theory and Applications*,*8*(3), 139-149. https://doi.org/10.1016/S0925-7721(96)00004-1