The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows.


Let us denote by $\Pi'\_\mathbb{R}$ the quotient of the free abelian group generated by the symbols $[P]$, where $P\subset \mathbb{R}^n$ is an arbitrary convex compact polytope, by the subgroup generated by elements
\begin{eqnarray\*}
(1)\, [P\cup Q]+[P\cap Q]-[P]-[Q] \mbox{ where } P,Q,P\cup Q \mbox{ are convex compact polytopes};\\
(2)\, [P+x]-[P] \mbox{ where } x\in \mathbb{R}^n.
\end{eqnarray\*}


Similarly let us define the analogous group $\Pi\_{\mathbb{Q}}'$ generated by convex compact polytopes with *rational* vertices and the same relations (1)-(2) (in the relation (2) one assumes $x\in\mathbb{Q}^n$).


**We have the obvious group homomorphism $\Pi\_{\mathbb{Q}}'\to \Pi'\_\mathbb{R}$. Is it injective?** A reference would be helpful.



The polytope algebras generated by polytopes with rational vs arbitrary vertices
Looking more carefully at the above McMullen's paper, I realized that the question has the positive answer due to Theorem 3 in the paper.


McMullen constructs homomorphisms $\Pi'\_{\mathbb{F}}\to \mathbb{F}$ (where $\mathbb{F}= \mathbb{R},\mathbb{Q}$) which all together separate points, i.e. if all of them vanish on some element, then the element vanishes. Let $u$ be a non-zero linear functional. For a polytope $P$ denote
$$P\_u:=\{x\in P|\, u(x)=\max\_{y\in P}u(y).\}$$
Let us define the homomorphism $f\_u\colon \Pi'\_{\mathbb{F}}\to \mathbb{F}$ on generators by $f\_u([P]):=vol(P\_u)$ where $vol$ is $(n-1)$-dimensional volume. McMullen shows that $f\_u$ is well defined.


This construction can be generalized recursively. Let $u\_1,\dots,u\_k$ be linearly independent linear functionals. Define $P\_{u\_1,\dots,u\_k}:=(P\_{u\_{1},\dots,u\_{k-1}})\_{u\_k}$. Define the homomorphism
$f\_{u\_1,\dots,u\_k}\colon \Pi'\_{\mathbb{F}}\to \mathbb{F}$ by
$$f\_{u\_1,\dots,u\_k}([P]):=vol(P\_{u\_{1},\dots,u\_{k}}),$$
where $vol$ is $(n-k)$-dimensional volume.


**Theorem 3.**(McMullen) All the homomorphisms $f\_{u\_1,\dots u\_k}$ separate points in
$ \Pi'\_{\mathbb{F}}$, i.e. if on some element $w\in \Pi'\_{\mathbb{F}} $ all such homomorphisms vanish then $w=0$.


Let us show how Theorem 3 answers (immediately) the question in the post. Let $w\in\Pi'\_{\mathbb{Q}} $ belongs to the kernel of the homomorphism in the question.
That implies that for all $k$-tuples of linearly $\mathbb{R}$-independent real linear functionals $u\_1,\dots,u\_k$ one has $f\_{u\_1,\dots,u\_k}(w)=0$. Now if $u\_1,\dots,u\_k$ are rational linear functionals which are linearly independent over $\mathbb{R}$ then they are linearly independent over $\mathbb{Q}$. For all of such $k$-tuples all $f\_{u\_1,\dots ,u\_k}$ separate points in $\Pi'\_{\mathbb{Q}}$. Hence $w=0$.


QED.


