Intersecting convex sets by rays
Radoslav Fulek, Andreas F. Holmsen, János Pach · 2008
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: