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.
Yes, these are well-studied. The creation of a polytope as the convex hull of the orbit of a point under a finite reflection group is known as Wythoff's construction, and the resulting polytopes are called "Wythoffian".
It is extensively discussed in the work of Coxeter, for instance in Regular Polytopes, his 1934 paper "Discrete Groups generated by Reflections" (DOI 10.2307/1968753), and his 1935 paper "Wythoff's Construction for Uniform Polytopes" (DOI 10.1112/plms/s2-38.1.327).
All Wythoffian polytopes are vertex-transitive, and the Wythoffian polytopes include all the polytopes whose reflection group acts transitively on the vertices (i.e. not the polytopes which are vertex-transitive only with rotations.) Almost all the known uniform polytopes are Wythoffian, and these are usually the ones of most interest.