### Abstract

A line transversal of a family S of n pairwise disjoint convex objects is a. straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in ℝ^{d} can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K(n ?1, d ?1) and K(n, d ?1), where K(n, d) Σ_{i=0}
^{d} (_{i}
^{n?1}) = Θ(n^{d}) is the number of pairs of antipodal cells in a simple arrangement of n great (d ?1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K(n, d ? 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations.

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

Pages (from-to) | 285-294 |

Number of pages | 10 |

Journal | Discrete and Computational Geometry |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - 2005 |

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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*,

*34*(2), 285-294. https://doi.org/10.1007/s00454-005-1174-2

**Geometric permutations induced by line transversals through a fixed point.** / Aronov, Boris; Smorodinsky, Shakhar.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 34, no. 2, pp. 285-294. https://doi.org/10.1007/s00454-005-1174-2

}

TY - JOUR

T1 - Geometric permutations induced by line transversals through a fixed point

AU - Aronov, Boris

AU - Smorodinsky, Shakhar

PY - 2005

Y1 - 2005

N2 - A line transversal of a family S of n pairwise disjoint convex objects is a. straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in ℝd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K(n ?1, d ?1) and K(n, d ?1), where K(n, d) Σi=0 d (i n?1) = Θ(nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d ?1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K(n, d ? 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations.

AB - A line transversal of a family S of n pairwise disjoint convex objects is a. straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in ℝd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K(n ?1, d ?1) and K(n, d ?1), where K(n, d) Σi=0 d (i n?1) = Θ(nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d ?1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K(n, d ? 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations.

UR - http://www.scopus.com/inward/record.url?scp=23944502302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23944502302&partnerID=8YFLogxK

U2 - 10.1007/s00454-005-1174-2

DO - 10.1007/s00454-005-1174-2

M3 - Article

VL - 34

SP - 285

EP - 294

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -