is $\{D\in{\mathscr{L}}:  C\leq D\}$. Geometrically it is the set of faces of ${\mathscr{L}}$ whose closure contains $C$. We will denote it by ${\mathscr{L}}_{\geq C}$.

\[lemma\_star\] Let $C\in{\mathscr{L}}$ and let $E(C)=\{e\in E :  C\subset H_{\alpha_e}\}$. Consider the hyperplane arrangement ${\mathcal{H}}(C)=(H_{\alpha_e})_{e\in E(C)}$ and let ${\mathscr{L}}_{{\mathcal{H}}(C)}$ be its face poset. Then the following three statements hold:

-   Each face $D$ of ${\mathscr{L}}$ is contained in a unique face $D'$ of ${\mathscr{L}}_{{\mathcal{H}}(C)}$, and the map $D \longmapsto D'$ induces an isomorphism of posets $\iota_C: {\mathscr{L}}_{\geq C} \stackrel{\sim}{\longrightarrow} {\mathscr{L}}_{{\mathcal{H}}(C)}$. In particular, it sends the chambers of ${\mathscr{T}}\cap{\mathscr{L}}_{\geq C}$ to the chambers of ${\mathscr{L}}_{{\mathcal{H}}(C)}$.

-   If $D_1,D_2\in{\mathscr{L}}_{\geq C}$ then the inclusion $S(D_1,D_2)\subset E(C)$ holds. In particular, we have the equality $S(D_1,D_2)=S(\iota_C(D_1),\iota_C(D_2))$, where the isomorphism $\iota_C$ is as in (i).

-   The isomorphism $\iota_C:{\mathscr{L}}_{\geq C} \stackrel{\sim}{\longrightarrow} 
    {\mathscr{L}}_{{\mathcal{H}}(C)}$ preserves composition and dimension, that is, for all $D,D'\in{\mathscr{L}}_{\geq C}$, the identities $\iota_C(D \circ D') = \iota_C(D)\circ\iota_C(D')$ and $\dim(\iota_C(D)) = \dim(D)$ hold.

In particular, ${\mathscr{L}}_{\geq C}$ is also isomorphic to the face poset of a regular cell decomposition of the unit sphere in $V/\bigcap_{e \in E(C)} H_{\alpha_e}$ that we denote by $\Sigma({\mathscr{L}}_{\geq C})$.

Statement (i) is clear.

We prove (ii). Let $D_1,D_2\in{\mathscr{L}}_{\geq C}$, and let $e\in S(D_1,D_2)$. Suppose for example 