Some aperture-angle optimization problems

Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P, the aperture angle of x with respect to Q is defined as the angle of the cone that: (1) contains Q, (2) has apex at x and (3) has its two rays emanating from x tangent to Q. We present algorithms with complexities O(n log in), O(n + n log (m/n)) and O(n + m) for computing the maximum aperture angle with respect