The layer number of α-evenly distributed point sets

For a finite point set in Rd, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number L(X) of a given point set X is defined as the number of steps of the peeling process in order to delete all points in X. It is known that if X is a set of random points in Rd, then the expectation of L(X) is Θ(|X|2∕(d+1)), and recently it was shown that if X is a point set of the square grid on the plane, then L(X)=Θ(|X|2∕3).