### Abstract

A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r > 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.

Original language | English (US) |
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Title of host publication | Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988 |

Publisher | Association for Computing Machinery, Inc |

Pages | 224-228 |

Number of pages | 5 |

ISBN (Electronic) | 0897912705, 9780897912709 |

DOIs | |

State | Published - Jan 6 1988 |

Event | 4th Annual Symposium on Computational Geometry, SCG 1988 - Urbana-Champaign, United States Duration: Jun 6 1988 → Jun 8 1988 |

### Publication series

Name | Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988 |
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### Other

Other | 4th Annual Symposium on Computational Geometry, SCG 1988 |
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Country | United States |

City | Urbana-Champaign |

Period | 6/6/88 → 6/8/88 |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

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## Cite this

*Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988*(pp. 224-228). (Proceedings of the 4th Annual Symposium on Computational Geometry, SCG 1988). Association for Computing Machinery, Inc. https://doi.org/10.1145/73393.73416