### Abstract

We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a /:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

Original language | English (US) |
---|---|

Pages (from-to) | 177-183 |

Number of pages | 7 |

Journal | Computational Geometry: Theory and Applications |

Volume | 21 |

Issue number | 3 |

State | Published - 2002 |

### Fingerprint

### Keywords

- -4-unbounded
- Convex sets
- Geometric transversal theory
- Helly-type theorem
- Jt-transversal

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*21*(3), 177-183.

**A Helly-type theorem for higher-dimensional transversals.** / Aronov, Boris; Goodman, Jacob E.; Pollack, Richard.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 21, no. 3, pp. 177-183.

}

TY - JOUR

T1 - A Helly-type theorem for higher-dimensional transversals

AU - Aronov, Boris

AU - Goodman, Jacob E.

AU - Pollack, Richard

PY - 2002

Y1 - 2002

N2 - We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a /:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

AB - We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a /:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

KW - -4-unbounded

KW - Convex sets

KW - Geometric transversal theory

KW - Helly-type theorem

KW - Jt-transversal

UR - http://www.scopus.com/inward/record.url?scp=33750727092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750727092&partnerID=8YFLogxK

M3 - Article

VL - 21

SP - 177

EP - 183

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 3

ER -