### Abstract

We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk^{5/3}), on the complexity of the kth level in an arrangement of n planes in ℝ^{3}, or on the number of k-sets in a set of n points in three dimensions, and we show that the complexity of the kth level in an arrangement of n line segments in the plane is O(n-√kα(n/k)), and that the complexity of the kth level in an arrangement of n triangles in 3-space is O(n^{2}k^{5/6}α(n/k)).

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

Pages (from-to) | 315-331 |

Number of pages | 17 |

Journal | Discrete and Computational Geometry |

Volume | 19 |

Issue number | 3 |

State | Published - 1998 |

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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*,

*19*(3), 315-331.

**On levels in arrangements of lines, segments, planes, and triangles.** / Agarwal, P. K.; Aronov, Boris; Chan, T. M.; Sharir, M.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 19, no. 3, pp. 315-331.

}

TY - JOUR

T1 - On levels in arrangements of lines, segments, planes, and triangles

AU - Agarwal, P. K.

AU - Aronov, Boris

AU - Chan, T. M.

AU - Sharir, M.

PY - 1998

Y1 - 1998

N2 - We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk5/3), on the complexity of the kth level in an arrangement of n planes in ℝ3, or on the number of k-sets in a set of n points in three dimensions, and we show that the complexity of the kth level in an arrangement of n line segments in the plane is O(n-√kα(n/k)), and that the complexity of the kth level in an arrangement of n triangles in 3-space is O(n2k5/6α(n/k)).

AB - We consider the problem of bounding the complexity of the kth level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk5/3), on the complexity of the kth level in an arrangement of n planes in ℝ3, or on the number of k-sets in a set of n points in three dimensions, and we show that the complexity of the kth level in an arrangement of n line segments in the plane is O(n-√kα(n/k)), and that the complexity of the kth level in an arrangement of n triangles in 3-space is O(n2k5/6α(n/k)).

UR - http://www.scopus.com/inward/record.url?scp=0032372467&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032372467&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032372467

VL - 19

SP - 315

EP - 331

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -