### Abstract

Consider an arrangement of n hyperplanes in R^{d}. Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no non-trivial bound was known for the general case where the polytopes may have overlapping interiors. We analyze families of polytopes that do not share vertices. In R^{3} we show an O(k^{1/3}n) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in R^{d} is Ω(k^{1/2}n^{d/2}) which is a factor of √n away from the best known upper bound in the range n^{d-2}≤k≤n^{d}.

Original language | English (US) |
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Title of host publication | Proceedings of the Annual Symposium on Computational Geometry |

Publisher | ACM |

Pages | 154-162 |

Number of pages | 9 |

State | Published - 1999 |

Event | Proceedings of the 1999 15th Annual Symposium on Computational Geometry - Miami Beach, FL, USA Duration: Jun 13 1999 → Jun 16 1999 |

### Other

Other | Proceedings of the 1999 15th Annual Symposium on Computational Geometry |
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City | Miami Beach, FL, USA |

Period | 6/13/99 → 6/16/99 |

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### ASJC Scopus subject areas

- Chemical Health and Safety
- Software
- Safety, Risk, Reliability and Quality
- Geometry and Topology

### Cite this

*Proceedings of the Annual Symposium on Computational Geometry*(pp. 154-162). ACM.