Separating convex sets in the plane

Jurek Czyzowicz, Eduardo Rivera-Campo, Jorge Urrutia, Joseph Zaks

Research output: Contribution to journalArticlepeer-review


Given a set A in R2 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 R2, 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 Hn and Gn, with n and 2 n disks, respectively, such that no pair of disks in Hn can be simultaneously separated from any set with more than one disk of Hn, and no disk in Gn can be separated from any subset of Gn with more than n disks. We also present a set Jm with 3 m line segments in R2, such that no segment in Jm can be separated from a subset of Jm 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 languageEnglish
Pages (from-to)189-195
Number of pages7
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - Dec 1992

ASJC Scopus subject areas

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


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