Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$.

Let $p_1,\dots,p_m$ be *all* lattice points in $P$.

**Question:** What is the condition on $P$ that guarantees that every lattice point in the dilation $nP$ can be expressed as $k_1p_1 + k_2p_2 + \cdots + k_n p_n$, where the $k_i$ are non-negative integers? Here, $n \in \mathbb{N}$
and $k_1+k_2+\cdots+k_n = n$.

*Note that not all $p_i$ need to be vertices of $P$.*
Clearly, all *vertices* of $nP$ are expressible in this manner, since they are dilations of the vertices in $P$.

*Remark:* The function $f(n)$ which counts lattice points in the dilation $nP$ is an (Erhart) polynomial and the $g(n)$ that counts the number of points that *can* be expressed as $k_1p_1 + k_2p_2 + \cdots + k_n p_n$ is *eventually* polynomial (Khovanskii).

Thus, we must impose some extra condition on the $p_i$s to have polynomiality all the way, and also equality.

Are there some non-trivial examples of such polytopes?