## Abstract

A set of objects is k-pierceable if there exists a set of k points such that each object is pierced by (contains) at least one of these points. Finding the smallest integer k such that a set is k-pierceable is NP-complete. In this paper, we present efficient algorithms for finding a piercing set (i.e., a set of k points as above) for several classes of convex objects and small values of k. In some of the cases, our algorithms imply known as well as new Helly-type theorems, thus adding to previous results of Danzer and Gruenbaum who studied the case of axis-parallel boxes. The problems studied here are related to the collection of optimization problems in which one seeks the smallest scaling factor of a centrally symmetric convex object K, so that a set of points can be covered by k congruent homothets of K.

Original language | English |
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Title of host publication | Proceedings of the 1996 12th Annual Symposium on Computational Geometry |

Pages | 113-121 |

Number of pages | 9 |

DOIs | |

State | Published - 1 Jan 1996 |

Externally published | Yes |

Event | Proceedings of the 1996 12th Annual Symposium on Computational Geometry - Philadelphia, PA, USA Duration: 24 May 1996 → 26 May 1996 |

### Conference

Conference | Proceedings of the 1996 12th Annual Symposium on Computational Geometry |
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City | Philadelphia, PA, USA |

Period | 24/05/96 → 26/05/96 |