Asymptotically optimal algorithm to compute if a line intersects a convex polygon

Question

An O(n) algorithm to detect if a line intersects a convex polygon consists in checking if any edge of the polygon intersects the line, and look if the number of intersections is

Answer 1

Bounding boxes may prove useful.

Recall that a convex part of an affine space is the intersection of all its envelope hyperplanes: you could get an approximation of your polygon (read a "bounding convex polygon") by considering only the intersection of a subset of the envelope hyperplanes, test for intersection of your line with this approximation, and refine if necessary.

All this works better if you expect a low number of intersections.