On Regular Vertices of the Union of Planar Convex Objects

AbstractLet $\mathcal{C}$ 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*(n4/3), 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 Cεf(n)⋅nε for any ε>0, where Cε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0).

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