Intersecting Convex Sets by Rays

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What is the smallest number tau = tau(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 tau 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 tau(n) when the convex bodies are fat and of roughly equal size.
Publisher
SPRINGER
Issue Date
2009-10
Language
English
Article Type
Article; Proceedings Paper
Keywords

DEPTH; POINTS

Citation

DISCRETE & COMPUTATIONAL GEOMETRY, v.42, no.3, pp.343 - 358

ISSN
0179-5376
URI
http://hdl.handle.net/10203/93616
Appears in Collection
MA-Journal Papers(저널논문)
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