## Abstract

Given a set A in R^{2} and a collection S of plane sets, we say that a line L separates A from S if A is contained in one of the closed half-planes defined by L, while every set in S is contained in the complementary closed half-plane. We prove that, for any collection F of n disjoint disks in R^{2}, there is a line L that separates a disk in F from a subcollection of F with at least {bottom right crop}(n-7)/4{bottom left crop} disks. We produce configurations H_{n} and G_{n}, with n and 2 n disks, respectively, such that no pair of disks in H_{n} can be simultaneously separated from any set with more than one disk of H_{n}, and no disk in G_{n} can be separated from any subset of G_{n} with more than n disks. We also present a set J_{m} with 3 m line segments in R^{2}, such that no segment in J_{m} can be separated from a subset of J_{m} with more than m+1 elements. This disproves a conjecture by N. Alon et al. Finally we show that if F is a set of n disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a line L that separates one of the segments from at least {bottom right crop}n/3{bottom left crop}+1 elements of F.

Original language | English |
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Pages (from-to) | 189-195 |

Number of pages | 7 |

Journal | Discrete and Computational Geometry |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1992 |

## ASJC Scopus subject areas

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