Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, you can assume that the convex region is described by intersection of closed half spaces? In the end, it will be great if you can generalize the problem to higher dimensions.
It so happens that Wikipedia contains an article entitled, "Intersection of a polyhedron with a line," but I doubt that answers your question.
A better answer is provided by another MO question, "Intersection points of straight line segment with Voronoi diagram": You can achieve $O(\log n)$ query time but only if you invest quadratic time preprocessing (which corroborates Ricky Demer's comment).