The layer number of grids

TL;DR

This paper investigates the peeling process of d-dimensional integer grids, establishing lower bounds on the number of steps needed and providing upper bounds for dimensions three and higher, advancing understanding of geometric peeling complexity.

Contribution

The paper extends previous 2D peeling analysis to higher dimensions, deriving new bounds on the layer number of d-dimensional integer grids.

Findings

Layer number of [n]^d is at least Ω(n^{2d/(d+1)}) for all d ≥ 1.

For d ≥ 3, the layer number is at most O(n^{d - 9/11}).

The approach enhances methods used in the 2D case to higher dimensions.

Abstract

The peeling process is defined as follows: starting with a finite point set $X \subset \mathbb{R}^d$, we repeatedly remove the set of vertices of the convex hull of the current set of points. The number of peeling steps needed to completely delete the set $X$ is called the layer number of $X$. In this paper, we study the layer number of the $d$-dimensional integer grid $[n]^d$. We prove that for every $d \geq 1$, the layer number of $[n]^d$ is at least $\Omega\left(n^\frac{2d}{d+1}\right)$. On the other hand, we show that for every $d\geq 3$, it takes at most $O(n^{d - 9/11})$ steps to fully remove $[n]^d$. Our approach is based on an enhancement of the method used by Har-Peled and Lidick\'{y} for solving the 2-dimensional case.