On the Boundary of the Union of Planar Convex Sets
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We give two alternative proofs leading to different generalizations of the following theorem of . Given n convex sets in the plane, such that the boundaries of each pair of sets cross at most twice, then the boundary of their union consists of at most 6n-12 arcs. (An arc is a connected piece of the boundary of one of the sets.) In the generalizations we allow pairs of boundaries to cross more than twice.