Counting convex polygons in planar point sets

Given a set S of n points in the plane, we compute in time O(n3) the total number of convex polygons whose vertices are a subset of S. We give an O(m · n3) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3,…, m; previously known bounds were exponential (O(nk/2). We also compute the number of empty convex polygons (resp.,k-gons, k m) with vertices in S in time O(n3) (resp., O(m · n3))