The layer number of $\alpha$-evenly distributed point sets

Abstract

For a finite point set in $\mathbb{R}^d$, 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 $\mathbb{R}^d$, then the expectation of $L(X)$ is $\Theta(|X|^{2/(d+1)})$, and recently it was shown that if $X$ is a point set of the square grid on the plane, then $L(X)=\Theta(|X|^{2/3})$. 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 $O(|X|^{3/4})$ 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 $O(|X|^{\frac{d+1}{2d}})$ for the layer number of an $\alpha$-evenly distributed point set $X$ in a unit ball in $\mathbb{R}^d$ for some $\alpha>1$ and $d\geq 3$.