Faster counting empty convex polygons in a planar point set

•Two improved algorithms that count empty convex polygons in a point set S are presented.•We can count all empty convex polygons in S in O(T) time, where T denotes the number of empty triangles in S.•We can count the number of empty convex k-gons in S in O(kT) time.•The expected number T of empty triangles is quadratic.