### Abstract

Let S be a finite collection of compact convex sets in ℝ^{d}. Let D(S) be the largest diameter of any member of S. We say that the collection 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 εD(S)/2 away from all d of the sets. We prove that if S is an ε-separated collection 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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Pages (from-to) | 507-517 |

Number of pages | 11 |

Journal | Discrete and Computational Geometry |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2001 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete and Computational Geometry*,

*25*(4), 507-517. https://doi.org/10.1007/s00454-001-0016-0