Intersecting Convex Sets by Rays

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 ({mathcal{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 (frac{dn+1}{d+1}) members of ({mathcal{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 (frac{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.