### 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) |
---|---|

Pages (from-to) | 507-517 |

Number of pages | 11 |

Journal | Discrete and Computational Geometry |

Volume | 25 |

Issue number | 4 |

State | Published - 2001 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

*25*(4), 507-517.

**A Helly-type theorem for hyperplane transversals to well-separated convex sets.** / Aronov, Boris; Goodman, J. E.; Pollack, R.; Wenger, R.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 25, no. 4, pp. 507-517.

}

TY - JOUR

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

AU - Aronov, Boris

AU - Goodman, J. E.

AU - Pollack, R.

AU - Wenger, R.

PY - 2001

Y1 - 2001

N2 - Let S be a finite collection of compact convex sets in ℝd. 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 ℝd 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 finite collection of compact convex sets in ℝd. 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 ℝd 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.

UR - http://www.scopus.com/inward/record.url?scp=0035617026&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035617026&partnerID=8YFLogxK

M3 - Article

VL - 25

SP - 507

EP - 517

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -