Lower bounds for intersection searching and fractional cascading in higher dimension

Given an n-edge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in O(k+ polylog(n)) time, using subquadratic storage? If the query is a plane and the input is a polytope with n vertices, can one achieve O(k+ polylog(n)) time with subcubie storage? Does any convex polytope have a boundary dominant Dobkin-Kirkpatrick hierarchy? Can fractional cascading be generalized to planar maps instead of linear lists? We prove that the answer to all of these questions is no, and we derive near-optimal solutions to these classical problems.