  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.
