Cutting disjoint disks by straight lines
Discrete & Computational Geometry
Lectures on Discrete Geometry
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Candle in the woods: asymptotic bounds on minimum blocking sets
Proceedings of the twenty-fifth annual symposium on Computational geometry
Centerpoints and Tverberg's technique
Computational Geometry: Theory and Applications
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What is the smallest number τ = τ(n) such that for any collection of n pairwise disjoint convex sets in d-dimensional Euclidean space, there is a point such that any ray (half-line) emanating from it meets at most τ sets of the collection? This question of Urrutia is closely related to the notion of regression depth introduced by Rousseeuw and Hubert (1996). We show the following: Given any collection C of n pairwise disjoint compact convex sets in d-dimensional Euclidean space, there exists a point p such that any ray emanating from p meets at most dn+1)/d+1) members of C. There exist collections of n pairwise disjoint (i) equal length segments or (ii) disks in the Euclidean plane such that from any point there is a ray that meets at least 2n/3--2 of them. We also determine the asymptotic behavior of τ(n) when the convex bodies are fat and of roughly equal size.