COUNTING CONVEX POLYGONS IN PLANAR POINT SETS

Given a set S of n points in the plane, we compute in time O(n(3)) the total number of convex polygons whose vertices are a subset of S. We give an O(m . n(3)) 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(n([k/2]))). We also compute the number of empty convex polygons (resp., k-gons, k less than or equal to m) with vertices in S in time O(n(3)) (resp., O(m . n(3))).