By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex *rational* polytopes.


If $P$ is an integral polytope, the counting function for the number of lattice points inside $nP$ is a polynomial $p(n)$.
This is the Erhart polynomial, and if $P$ does not have integral vertices,
$p(n)$ is in general just a quasi-polynomial.


A integrally closed polytope is a polytope, such that
each lattice point in $nP$ is a sum of exactly $n$ lattice points in $P$.


**To the question:**
Now, let $P$ be a *non-integral* polytope, but its Erhart quasi-polynomial *is* in fact a polynomial. Let $P'$ be the convex hull of the lattice points in $P'$.


Are there examples $P$ for which $P'$ is *not* integrally closed?


**Motivation:** Being integrally closed is a quite rare property for a polytope (in high dimensions). Having a polynomial Erhart function, when only a quasi-polynomial is expected is also rare. Therefore, the chance that these coincide is even rarer, so one would expect a lot of examples above, or these properties are related somehow.



Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials
A combination of the polytopes introduced in Exercise 2.24 of [*Polytopes, Rings, and K-Theory*](http://math.sfsu.edu/gubeladze/publications/kripo.html) by W. Bruns and J. Gubeladze (see also Section 3 of [this paper](http://arxiv.org/abs/1307.7422)) with a 3-dimensional version of the polygons in [this paper by T. McAllister and K. Woods](http://arxiv.org/abs/math/0310255) should do the trick. To be more concrete, I believe the convex hull of $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(0, \frac 1 a - 1, 1)$, $(1, \frac 1 a - 1, 1)$, $(0,0,a)$, $(1,0,a)$, $(0,1,a)$, $(1,1,b)$, $(1,1,a+b)$ will give you an example with the right choices of $a$ and $b$ (the integer $b$ has to be chosen large enough so that the polytope is not normal---try something like $b=4$---, and $a$ has to be chosen large enough so that these 10 points are actually vertices).


