Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
The maximum number of ways to stab n convex nonintersecting sets in the plane is 2n - 2
Discrete & Computational Geometry
Discrete & Computational Geometry
The different ways of stabbing disjoint convex sets
Discrete & Computational Geometry
Shape sensitive geometric permutations
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
On neighbors in geometric permutations
Discrete Mathematics
Geometric permutations of balls with bounded size disparity
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
Geometric permutations of disjoint unit spheres
Computational Geometry: Theory and Applications
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A line transversal of a family S of n pairwise disjoint convex objects is a straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other.In this note we consider a long-standing open problem in transversal theory, namely that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in Rd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n 2 pairwise disjoint convex objects that are induced by lines passing through any fixed point is between K(n - 1, d - 1) and K(n,d - 1), where K(n,d) = Σdi=0 (n-1/i) = Θ(nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d - 1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all lines transversal through a fixed point to a family of n 2 possibly intersecting convex objects is K(n, d - 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations.