The Strong Bruhat Order of a finite irreducible Coxeter Group satisfies all the axioms of being an abstract polytope. It's also a remarkably nice fact that the Weak Bruhat Order of the Symmetric Group corresponds to the nodes and edges of a permutahedron. Is something analogous true for all finite irreducible Coxeter Groups?
More specifically; is the Weak Order (equivalently, the Cayley graph) of a finite irreducible Coxeter Group the 1-skeleton of some 'nice/interesting' polytope? And if so, what's the relationship between the polytope of the Strong Bruhat Order and the polytope of the Weak Order?
I suspect this is well-known and perhaps more generally true in a wider context! But I can't seem to find the answer easily for myself!
If $G$ is a finite Coxeter group, let $\rho:G\to\mathrm{O}(\Bbb R^d)$ be its represenation as a reflection group $W:=\rho(G)$. Choose a generic point $x\in\Bbb R^d$, that is, $x$ is not fixed by $\rho(g)$ for any $g\in G$ other than the identity. Consider the polytope
$$P:=\mathrm{conv}\{\rho(g)x\mid g\in G\}.$$
This is called an orbit polytope of $W$ with generator $x$, and since $W$ is a reflection group and $x$ is generic, this is also called a $W$-permutahedron.
As it turns out, the Cayley graph of $G$ is isomorphic to the edge-graph (aka. 1-skeleton) of this polytope.