Consider a finite reflection group acting on $\mathbb{R}^N$. Pick a vector $x \in \mathbb{R}^N$ and look at the convex hull of its orbit. This is a polytope.
There are some interesting particular cases. For example, the unit $l_1$ ball can be seen as the convex hull of the orbit of the vector $(1, 0, \ldots, 0)$ under the action of the group of signed permutations. Likewise, the $l_\infty$ can be seen as the convex hull of the orbit of the vector $(1, 1, \ldots, 1)$ under the same action.
Are these polytopes well studied? What can be said about them? I would appreciate some references.