On intersecting a set of parallel line segments with a convex polygon of minimum area

Let S = {l1, l2, l3, . . . , ln} be a set of n vertical line segments in the plane. Though not essential, to simplify proofs we assume that no two li’s are on the same vertical line. A convex polygon weakly intersects S if it contains a point of each line segment on its boundary or interior. In this paper, we propose an O(n log n) algorithm for the problem of finding a minimum area convex polygon that weakly intersects S.