Explicit bounds for the layer number of the grid

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 $d$.