Before revision:  We prove that when X consists of points independently and uniformly sampled inside a convex polygon with k vertices , the expected number of vertices on the first t convex shells is O(kt\log{n\left(\right) and the expected number of vertices on the first t convex layers is O\left(kt^{3} \log{\frac{n \right).
After revision:  We prove that when X consists of points independently and uniformly sampled inside a convex polygon with k vertices , the expected number of vertices on the first t Tukey layers is O\left(\right) and the expected number of vertices on the first t convex layers is O\left(kt^{3} \log{\frac{n \right).