Convex peels of a random point set.
Preprint

The convex hull peels of a finite point set in R^d are of appeal in multivariate statistics and impose an intuitive order onto the points. These convex layers are consecutively obtained by iterating the two operations: (1) construct the convex hull of the existing point set and (2) remove the vertices of the convex hull from the point set. This note shows that the number N_n,i of vertices of the i-th convex hull peel (for i \geq 2 not too large) of a uniform sample of size n from a convex polygon or a circular disk has asymptotic (in n) expectation ((2/3) log n)^{i -1} E N_n,1 and variance ((2/3) log n)^{2(i -1)} Var ( N_n,1 ).