On regular vertices of the union of planar convex objects

Esther Ezra*, János Pach, Micha Sharir

*Corresponding author for this work

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


FundersFunder number
Israel Science Fund
National Science FoundationCCF-05-14079
International Business Machines Corporation
United States-Israel Binational Science Foundation155/05
Tel Aviv University


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


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