Geometric permutations of disjoint translates of convex sets
Discrete Mathematics
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Discrete & Computational Geometry
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Discrete & Computational Geometry
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
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On neighbors in geometric permutations
Discrete Mathematics
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SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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We prove that a set of n unit balls in Rd admits at most four distinct geometric permutations, or line transversals, thus settling a long-standing conjecture in combinatorial geometry. The constant bound significantly improves upon the &THgr;(nd-1) bound for the balls of arbitrary radii. Intrigued by this large gap between the two bounds, we also investigate how the number of geometric permutations varies as a function of shape, size, and spacing of objects. Our results include a tight bound of 2d-1 on the geometric permutations of n disjoint rectangular boxes in Rd, and a constant bound on the geometric permutations for disks in the plane when the ratio between the largest to smallest disks is bounded. An important consequence of the former theorem is that if the smallest bounding boxes containing a set of geometric objects in Rd are pairwise disjoint, then those objects admit only 2d-1 permutations, which is a significant improvement on the &Ogr;(n2d-2) bound known for general convex objects.