Before revision: Given a planar point set X, we study the convex shells and the convex layers . 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 study the Tukey layers and convex layers of a planar point set, which consists of n 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).