On Regular Vertices of the Union of Planar Convex Objects

Let be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in intersect in at most s points for some constant s≥4. 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 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). Work by János Pach and Micha Sharir was supported by NSF Grant CCF-05-14079, and by a grant from the U.S.–Israeli Binational Science Foundation. Work by Esther Ezra and Micha Sharir was supported by grant 155/05 from the Israel Science Fund and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the first author has also been supported by an IBM Doctoral Fellowship. A preliminary version of this paper has been presented in Proc. 23nd Annu. ACM Sympos. Comput. Geom., 2007, pp. 220–226. E. Ezra’s current address: Department of Computer Science, Duke University, Durham, NC 27708-0129, USA. E-mail: esther@cs.duke.edu