### Abstract

Let S be a family of compact convex sets in R^{d}. Let D(S) be the largest diameter of any member of S. The family S is ε-separated if, for every 0<k<d, any k of the sets can be separated from any other d-k of the sets by a hyperplane more than ε/D(S) away from all d of the sets. We prove that if S is an ε-separated family of at least N(ε) compact convex sets in R^{d} and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N(ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

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

Publisher | ACM |

Pages | 57-63 |

Number of pages | 7 |

State | Published - 2000 |

Event | 16th Annual Symposium on Computational Geometry - Hong Kong, Hong Kong Duration: Jun 12 2000 → Jun 14 2000 |

### Other

Other | 16th Annual Symposium on Computational Geometry |
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City | Hong Kong, Hong Kong |

Period | 6/12/00 → 6/14/00 |

### Fingerprint

### 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. 57-63). ACM.

**Helly-type theorem for hyperplane transversals to well-separated convex sets.** / Aronov, Boris; Goodman, Jacob E.; Pollack, Richard; Wenger, Rephael.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Annual Symposium on Computational Geometry.*ACM, pp. 57-63, 16th Annual Symposium on Computational Geometry, Hong Kong, Hong Kong, 6/12/00.

}

TY - GEN

T1 - Helly-type theorem for hyperplane transversals to well-separated convex sets

AU - Aronov, Boris

AU - Goodman, Jacob E.

AU - Pollack, Richard

AU - Wenger, Rephael

PY - 2000

Y1 - 2000

N2 - Let S be a family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family S is ε-separated if, for every 0d and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N(ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

AB - Let S be a family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family S is ε-separated if, for every 0d and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S. The number N(ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

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UR - http://www.scopus.com/inward/citedby.url?scp=0033707561&partnerID=8YFLogxK

M3 - Conference contribution

SP - 57

EP - 63

BT - Proceedings of the Annual Symposium on Computational Geometry

PB - ACM

ER -