of the hyperplane $H_{\alpha_e}$.

Fix a chamber $B\in{\mathscr{T}}$. We can then define a partial order $\preceq_B$ on ${\mathscr{T}}$ by declaring that $T\preceq_B T'$ if and only if $S(B,T) \subseteq S(B,T')$. The resulting poset is called the *chamber poset with base chamber $B$*. We will denote it by ${\mathscr{T}}_B$. It is a poset with minimal element $B$ and maximal element $-B$. When all the hyperplanes are distinct, this poset is also graded with the poset rank function $\rho(T) = |S(B,T)|$; see [@OM Proposition 4.2.10].

We also consider the graph with vertex set ${\mathscr{T}}$, where two chambers $T,T'\in{\mathscr{T}}$ are connected by an edge if and only if $\overline{T}\cap\overline{T}'$ spans a hyperplane (necessarily one of the $H_{\alpha_e}$). In this situation, we say that this hyperplane is a *wall* of the chambers $T$ and $T'$. This graph is called the *chamber graph*. In the case when all the hyperplanes of the arrangement ${\mathcal{H}}$ are distinct, the distance between two chambers $T$ and $T'$ in this graph is $|S(T,T')|$; see [@OM Proposition 4.2.3].

Consider the sphere $\mathbb{S}$ of center $0$ and radius $1$ in $V/V_0$. The intersections $\overline{C}\cap\mathbb{S}$, for $C\in{\mathscr{L}}$, form a regular cell decomposition $\Sigma({\mathscr{L}})$ of $\mathbb{S}$, and we will identify ${\mathscr{L}}$ with the 