Suppose that $G\subset O(d)$ is a finite reflection (finite Coxeter) group. For any $v\in \mathbb{R}^d$ which is not fixed by any non-trivial $g\in G$, one can consider the orbit polytope (Coxeter) permutahedra \begin{equation} P(G;v)=Conv (G\cdot v) \end{equation} given by the orbit.

Now consider $G^+\subset SO(d)$, the index-two rotation subgroup of $G$. Again one can consider the orbit polytope \begin{equation} P(G^+;v)=Conv(G^+\cdot v) \end{equation} for $v$ as above (i.e., not fixed by any non-trivial $g$ from the original group). Is it necessarily the case that $P(G^+;v)$ is just obtained by $P(G;v)$ by "alternation"?

If $v_1,v_2\in \mathbb{R}^d$ are not fixed by any $g\in G$, it can be shown that $P(G;v_1)$ and $P(G;v_2)$ are combinatorially equivalent. Must the same be true for $P(G^+;v_1)$ and $P(G^+;v_2)$ as well?

This would definitely seem to be the case for a number of examples (e.g., for $G=A_2\times A_2\times A_2$, for which $P(G;v)$ is a box, $P(G^+;v)$ is a tetrahedra).

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