Abstract.
We give two alternative proofs leading to different generalizations of the following theorem of [1]. 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.
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Received May 5, 1997, and in revised form July 15, 1997.
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Pach, J., Sharir, M. On the Boundary of the Union of Planar Convex Sets . Discrete Comput Geom 21, 321–328 (1999). https://doi.org/10.1007/PL00009424
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DOI: https://doi.org/10.1007/PL00009424