Let $A$ be a matrix with entries either 0 or 1, where each column contains at least one 1, to remove trivial degenerations.

Let $P$ be the *convex hull of all integer vectors* $x$ that satisfy $Ax \leq y$, and $x\geq 0$, where $y$ is some non-negative integer vector. Clearly, $P$ is an integral polytope.

For example (to address David Speyers comment), when $$A=\begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}, y=(1,1,1)$$ then $P$ is the convex hull of the solutions to $Ax\leq y$, so $P$ is the convex hull of $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$, the standard simplex.

Doing some computer experiments, I believe the following:

*Conjecture:* P is integrally closed, i.e., every *integer* point $p \in kP$
can be expressed as $p=p_1+p_2+\dots+p_k$ where all $p_i$ are *integer* points in $P$,
whenever $k$ is a natural number.

In the example above, this is known to be integrally closed.

Note that there are no conditions on the minors of $A$.

Is this a known result? This seems hard, since we do not have a nice description of $P$, that is, the supporting hyperplanes, nor the vertices, are explicitly known.