For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.


**Question:** Given $d > n + 2$ is it true that


$$ dP \cap \mathbb{Z}^n \stackrel{?}{=} (n+2)P \cap \mathbb{Z}^n + (d-(n+2))P \cap \mathbb{Z}^n \ ? $$


That is for any integer point $u$ in the dilation $dP$ is there an integer point $v$ in the dilation $(n+2)P$ such that $u-v$ is an integer point of the dilation $(d-(n+2))P$? If so, is the construction explicit? If not are there conditions on $P$ where it does hold?



Integer decomposition of dilated integral polytopes
The answer is YES in the following stronger form:


**Lemma 1:** Let $d>n$ and let $p\in dP\cap \mathbb{Z}^n$. Then there is an $i\in\{1,\dots, n\}$ and points $p'\in iP\cap \mathbb{Z}^n$, $p\_1,\dots, p\_{d-i}\in P\cap \mathbb{Z}^n$ such that
$$
p = p' + p\_1 + \cdots + p\_{d-i}.
$$


For the proof I will (as is customary in this type of problems) work on the cone over $P$,
defined as follows:
$$
C\_P := \operatorname{pos}(P\times \{1\}) \subset \mathbb{R}^{n+1}.
$$
Observe that $dP$ can be identified with $C\_P \cap \{x\_{n+1}=d\}$.
I call height of a point in $C\_P$ the value of its last coordinate, so that integer points at height $d$ are identified with integer points in $dP$.
Lemma 1 is equivalent to:


**Lemma 2:** Let $p\in C\_P\cap \mathbb{Z}^{n+1}$ be an integer point at height $d> n$. Then, there is a point $p'\in C\_P$ at a certain height $i\in \{1,\dots,n\}$ and integer points points $p\_1,\dots, p\_{d-i}$ such that $p=p' + p\_1 + \cdots + p\_{d-i}$.


*Proof of Lemma 2*:
There is no loss of generality in assuming that $P$ is a simplex (if not, consider an integer triangulation of $P$ and apply the result to each individual simplex). So, let $v\_1,\dots,v\_{n+1}$ be the vertices of $P$, considered as points in $C\_P$ at height one, and let $Z$ be the unit parallelepiped obtained as the Minkowski sum of the segments $Ov\_i$. 


The cone $C\_P$ can be tiled by integer translations of $Z$; more precisely, it is tiled by all translations $v+Z$ where $v$ runs over the integer non-negative combinations of the $v\_i$'s. Thus, $p$ can be decomposed as $p'+v$, where $p'$ is an integer point in $Z$ and $v$ is an integer non-negative combination of the $v\_i$'s. 


Except in the trivial cases where $p'=O$ or $p'$ is the sum of all the $v\_i$'s, the integer point $p'$ in $Z$ has height between $1$ and $n$, which finishes the proof.
QED


