Explicit bounds for the layer number of the grid

TL;DR

This paper establishes explicit upper and lower bounds for the layer number of a grid, showing it grows quadratically with the grid size and linearly with the dimension, improving previous bounds.

Contribution

It provides a simplified proof for an improved upper bound and establishes a new linear lower bound for the grid's layer number.

Findings

Upper bound of (1/4)dn^2 + 1 for the layer number

Lower bound of (1/2)d(n-1) + 1 for the layer number

Layer number grows quadratically with grid size and linearly with dimension

Abstract

The number of steps required to exhaust a point set by iteratively removing the vertices of its convex hull is called the layer number of the point set. This article presents a short proof that the layer number of the grid $\{1,2,\dots,n\}^d$ is at most $\frac{1}{4}dn^2+1$, significantly improving the dependence on $d$ in the best-known upper bound. We also prove a lower bound of $\frac{1}{2}d(n-1)+1$, which shows that the layer number of the grid is linear in $d$.