**Update**

I recently uploaded a preprint in which I work out the details that are missing below.
So in fact, vertex-transitive zonotopes are $\Gamma$-permutahedra.

I believe to have (at least a roadmap to) a proof of the following:

**Theorem.** If $P\subset\Bbb R^d$ is a vertex-transitive zonotope, then $P$ is a $\Gamma$-permutahedron. That is, $P$ is the convex hull of the orbit of an appropriately chosen point $\smash{v\in\Bbb R^d}$ under a finite reflection group $\smash{\Gamma\subset\mathrm{GL}(\Bbb R^d)}$.

In other words, $P$ is the omnitruncation of some uniform polytope (when considered with a certain subgroup of its symmetries).

I will give some thoughts about my proof, since I have not thought through every detail:

- Every zonotope can be uniquely written as the Minkowski sum of line segments with pair-wise trivial intersection.
- Let's call $r\in\Bbb R$ a root of $P$ if $\mathrm{conv}\{-r,r\}$ is one of these line segments.
- One then shows that the set of roots of $P$ forms a
*root system* (without integrality condition).^{1}
- One further shows, that the zonotope $P$ has the same symmetries as its set of roots, hence that its symmetry group is a reflection group.

(until here, I think, David had another approach using the normal fan of $P$).

- Let $\tilde \Gamma$ be the symmetry group of $P$. Since $P$ is vertex-transitive, $P$ is the orbit polytope of some point $\smash{v\in\Bbb R^d}$ w.r.t $\smash{\tilde \Gamma}$. As David observed, this group might be too large to call $P$ a $\smash{\tilde\Gamma}$-permutahedron.
- Consider the subgroup $\Gamma\subseteq\tilde\Gamma$ generated by all reflections in $\tilde\Gamma$ that fix no vertex of $P$. Then $\Gamma$ is a reflection group.
- Show that $P$ is the orbit polytope of $v$ under $\Gamma$. Then $\Gamma$ acts vertex-transitively and -regularly on $P$, hence $P$ is a $\Gamma$-permutahedron.

**Some notes on **^{1}

Let $R$ be the set of roots of $P$. How to show that $R$ is a root system:

- Choose any two (linearly independent) $r,r'\in R$ and consider the 2-dimensional set $R':=\mathrm{span}\{r,r'\}\cap R$.
- Let $P'$ be the zonotope generated by $R'$. This zonotope is a 2-face of $P$, and by using the argument that $\mathrm{Aut}(P)=\mathrm{Aut}(R)$ one can conclude that from the vertex-transitivity of $P$ follows the vertex-transitivity of $P'$. (This part is sketchy right now, and makes some trouble. How to fix this? I think that that the faces of a vertex-transitive polytope do not necessarily have to be vertex transitive!
*Update*: yes they are vertex-transitive, see the preprint)
- It follows that $P'$ is a $2n$-gon with possibly alternating edge lengths.
- One convinces oneself that the roots of $P'$ are a root system ($2n$ roots equally spaces by $\pi/n$, maybe of alternating lengths), that is, $R'$ (and hence $R$) contains the reflection of $r'$ on the hyperplane defined by $r$.
- Since $r$ and $r'$ were chosen arbitrarily, this shows that $R$ is a root system.

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