The polytope algebras generated by polytopes with rational vs arbitrary vertices

Question

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.

Answer 1

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.