The layer number of alpha-evenly distributed point sets

Abstract

For a finite point set in R-d, 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 R-d, then the expectation of L(X) is Theta(vertical bar X vertical bar(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) = Theta(vertical bar X vertical bar(2/3)). In this paper, we investigate the layer number of alpha-evenly distributed point sets for alpha > 1; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an alpha-evenly distributed point set for some alpha > 1. We find an upper bound of O(vertical bar X vertical bar(3/4)) for the layer number of an alpha-evenly distributed point set X in a unit disk on the plane for some alpha > 1, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of O(vertical bar X vertical bar(d+1/2d)) for the layer number of an a-evenly distributed point set X in a unit ball in R-d for some alpha > 1 and d >= 3.