**Background**: Let $W$ be a finite reflection group of rank $n$, acting on $\mathbb{R}^n$. The reflecting hyperplanes of $W$ meet the unit sphere $S^{n-1}\subset\mathbb{R}^n$, inducing a simplicial complex structure on $S^{n-1}$ ("Coxeter complex of $W$"). $W$ acts freely and transitively on the facets of this triangulation ("the chambers"). Fixing a specific chamber, the $n$ reflections in the $(n-1)$-faces ("walls") of this chamber are "Coxeter generators" and one gets a metric on the chambers: given chambers $P,Q$, find the unique $w\in W$ with $wP=Q$ and define $d(P,Q)$ to be the minimum length of the word in the Coxeter generators required to represent $w$. The Cayley graph of $W$ with respect to the Coxeter generators can be identified with the graph with vertices and edges the chambers and walls of the Coxeter complex; then this metric is the graph metric.

One also gets a metric by picking a distinguished point of each chamber and then just taking the distance between chambers $P$ and $Q$ to be the Riemannian distance in the sphere $S^{n-1}$. To pin down this possibility, let us use the barycenter of each chamber as its distinguished point.

Question: How do these two metrics relate?

It seems to me that they should be pretty closely related. The Coxeter complex is a very regular object. For example, $S_4$ acting in its standard representation on $\mathbb{R}^3$ is such a Coxeter group; the barycenters of the coxeter complex form the vertices of a truncated octahedron, embedded in $S^2$. The Cayley-graph distance is the path-length of the shortest path between vertices. Vertices further away on the sphere $S^2$ also require longer paths.

But can this be made precise? E.g. are there universal positive constants $A,\varepsilon$ with $\varepsilon$ reasonably small so that $A(1-\varepsilon)d_{graph}(P,Q)\leq d_{Riemannian}(P,Q)\leq A(1+\varepsilon)d_{graph}(P,Q)$? Or are there such constants that depend on $n$ in an explicitly controlled way?

**Motivation**: The present question follows up an attempt I made to answer this question. I posted a partial answer with the uncertainty in my answer coming precisely from my uncertainty about the present question.