The layer number of [formula omitted]-evenly distributed point sets

Abstract

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).In this paper, we investigate the layer number of α-evenly distributed point sets for α>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 α-evenly distributed point set for some α>1. We find an upper bound of O(|X|3∕4) for the layer number of an α-evenly distributed point set X in a unit disk on the plane for some α>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(|X|d+12d) for the layer number of an α-evenly distributed point set X in a unit ball in Rd for some α>1 and d≥3.