Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is
$$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \Gamma\}.$$
I always wondered whether there is an easy way to decide whether two vertices of the orbit polytope define an edge. More precisely, I wonder the following:
Question: Given vertices $v,w$ of $P$, is it true that $\mathrm{conv}\{v,w\}$ is an edge of $P$ if and only if the linear functional $\langle v+w,\cdot\rangle$ (as map on the vertices of $P$) is maximized exactly on $v$ and $w$?
This can fail for inscribed polytopes, so the symmetry might be relevant. I suspect that this is too naive, but I found no counterexample.
