On regular vertices of the union of planar convex objects

Esther Ezra, János Pach, Micha Sharir

Research output: Contribution to journalArticlepeer-review


Let C be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in C intersect in at most s points for some constant s4. We show that the maximum number of regular vertices (intersection points of two boundaries that intersect twice) on the boundary of the union U of C is O*(n 4/3), which improves earlier bounds due to Aronov et al. (Discrete Comput. Geom. 25, 203-220, 2001). The bound is nearly tight in the worst case. In this paper, a bound of the form O* (f(n)) means that the actual bound is C ε f(n) n ε for any ε>0, where C ε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0).

Original languageEnglish
Pages (from-to)216-231
Number of pages16
JournalDiscrete and Computational Geometry
Issue number2
StatePublished - Mar 2009


  • (1/r)-cuttings
  • Bi-clique decompositions
  • Geometric arrangements
  • Lower envelopes
  • Regular vertices
  • Union of planar regions


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