Intersecting convex sets by rays

What is the smallest number τ = τ(<i>n</i>) such that for any collection of <i>n</i> pairwise disjoint convex sets in <i>d</i>-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 <i>C</i> of <i>n</i> pairwise disjoint compact convex sets in <i>d</i>-dimensional Euclidean space, there exists a point <i>p</i> such that any ray emanating from <i>p</i> meets at most <i>dn</i>+1)/<i>d</i>+1) members of <i>C</i>. 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 2<i>n</i>/3--2 of them. We also determine the asymptotic behavior of τ(<i>n</i>) when the convex bodies are fat and of roughly equal size.

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