Intersecting Convex Sets by Rays

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.