### Abstract

Color red and blue the n vertices of a convex polytope P in ℝ^{3}. Can we compute the convex hull of each color class in o(n log n)? What if we have x > 2 colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eight-year old open problem by showing how to split a convex polytope in linear expected time.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09 |

Pages | 61-70 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 4 2009 |

Event | 25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark Duration: Jun 8 2009 → Jun 10 2009 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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### Other

Other | 25th Annual Symposium on Computational Geometry, SCG'09 |
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Country | Denmark |

City | Aarhus |

Period | 6/8/09 → 6/10/09 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Keywords

- Convex polytope
- Half-space range searching
- Hereditary convex hulls

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## Cite this

*Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09*(pp. 61-70). (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1542362.1542374