New estimates for convex layer numbers

TL;DR

This paper investigates the layer number of evenly distributed point sets within the unit ball, establishing new bounds that improve previous results and providing constructions that nearly match these bounds.

Contribution

It introduces improved asymptotic bounds for the convex layer number of evenly distributed point sets in high dimensions, including near-tight constructions.

Findings

Lower bound: $L(X) \\geq \\Omega(|X|^{1/d})$

Upper bound: $L(X) \\leq O(|X|^{2/d})$ for $d \\geq 2$

Nearly tight constructions with $L(X) = \\Theta(|X|^{2/d - 1/(d 2^{d-1})})$

Abstract

Starting with a finite point set $X \subset \mathbf{R}^d$, the peeling process repeatedly removes the set of the vertices of the convex hull of the current set. The number of peeling steps required to completely remove $X$ is called the layer number of $X$, denoted by $L(X)$. In the article, we study the layer number of evenly distributed families of point sets contained in $B^d$, the $d$-dimensional unit ball. These sets consist of points in $B^d$ whose minimal distance is asymptotically as large as possible. We show that for a set $X$ belonging to an evenly distributed family, $L(X) \geq \Omega(|X|^{1/d})$ holds, with the bound being asymptotically sharp. On the other hand, building on earlier results, we prove that $L(X)\leq O(|X|^{2/d})$ holds for $d\geq 2$, which improves greatly on the current upper bound of $O(|X|^{(d+1)/2d})$ for $d \geq 3$. Finally, we provide a recursive construction of evenly distributed families whose sets satisfy $L(X) = \Theta(|X|^{2/d - 1/(d 2^{d-1})})$, showing that our upper bound is nearly tight.