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On the boundary of the union of planar convex sets. (English) Zbl 0927.52001
Let $$\mathcal C$$ be a collection of $$n \geq 3$$ nondegenerate convex sets in the plane, any two of which have at most a finite number of boundary points in common. If two members of $$\mathcal C$$ have exactly two boundary points in common, then these points are called regular vertices of the arrangement $$\mathcal A(\mathcal C).$$ All other intersection points of the boundary curves are said to be irregular. Let $$U =\cup \mathcal C$$ denote the union of all members of $$\mathcal C.$$ Let $$R$$ and $$I$$ denote the set of regular and irregular vertices of $$\mathcal A(\mathcal C),$$ respectively, lying on $$\partial U,$$ the boundary of $$U.$$ Further, put $$V= R\cup I.$$ If the sets in $$\mathcal C$$ are bounded, then $$| V|$$ is equal to the number of arcs that compose $$\partial U.$$
It was shown in K. Kedem, R. Livne, J. Pach and M. Sharir, Discrete Comput. Geom. 1, 59-71 (1986; Zbl 0594.52004) that if any two members of $$\mathcal C$$ have at most two boundary points in common, then $$| R|= | V|\leq 6n - 12,$$ and this bound is tight in the worst case. The authors generalize this result as follows: For any collection of $$n\geq 3$$ nondegenerate convex sets in general position in the plane satisfying the above assumptions, the bound $$| R|\leq 2| I|+6n-12$$ holds.

##### MSC:
 52A10 Convex sets in $$2$$ dimensions (including convex curves) 52A40 Inequalities and extremum problems involving convexity in convex geometry
Zbl 0594.52004
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