Counting Convex k-gons in an Arrangement of Line Segments

Let A(S) be the arrangement formed by a set of n line segments S in the plane. A subset of arrangement vertices p1, p2, . . . , pk is called a convex k-gon of A(S) if (p1, p2, . . . , pk) forms a convex polygon and each of its sides, namely, (pi, pi+1) is part of an input segment. We want to count the number of distinct convex k-gons in the arrangement A(S), of which there can be Θ(n) in the worst-case. We present an O(n log n+mn) time algorithm, for any fixed constant k, where m is the number of pairwise segment intersections. We can also report all the convex k-gons in time O(n log n+mn+|K|), where K is the output set. We also prove that the k-gon counting problem is 3SUM-hard for k = 3 and k = 4.