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Let $\mathcal{C}$be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in $\mathcal{C}$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 $\mathcal{C}$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 ε0, where C ε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0).

On Regular Vertices of the Union of Planar Convex Objects