Proposition 4

Let $C$ be a chamber and $\mathfrak{N}$ a set of walls of $C$. Let $W_{\mathfrak{N}}$ be the subgroup of $W$ generated by the orthogonal reflections with respect to the elements of $\mathfrak{N}$. For $H \in \mathfrak{N}$, denote by $e_H$ the unit vector orthogonal to $H$ on the same side of $H$ as $C$. The following conditions are equivalent: (i) The group $W_{\mathfrak{N}}$ is finite. (ii) There exists a point of $E$ invariant under every element of $W_{\mathfrak{N}}$. (iii) The hyperplanes belonging to $\mathfrak{N}$ have non-empty intersection. (iv) The family of vectors $(e_H)_{H \in \mathfrak{N}}$ is free in $T$.

By property (D2) at the beginning of § 3, the stabiliser in $W$ of every point of $E$ is finite, so (ii) implies (i).

Since the group $W_{\mathfrak{N}}$ is generated by the set of reflections with respect to the hyperplanes belonging to $\mathfrak{N}$, the fixed points of $W_{\mathfrak{N}}$ are the points of $E$ belonging to every hyperplane $H \in \mathfrak{N}$, hence the equivalence of (ii) and (iii).

Assume that there exists a point $a$ of $E$ such that $a \in H$ for all $H \in \mathfrak{N}$ and let $t \in T$ be such that $a + t \in C$. Since $(e_H|e_{H'}) \leq 0$ for $H, H' \in \mathfrak{N}$ such that $H \neq H'$ (Prop. 3), and since $(t|e_H) > 0$ for all $H \in \mathfrak{N}$, Lemma 3 (ii) implies that the $e_H$ for $H \in \mathfrak{N}$ are linearly independent. Consequently, (iii) implies (iv).

Suppose finally that the family $(e_H)_{H \in \mathfrak{N}}$ is free. Let $a$ be a point of $E$. For any hyperplane $H \in \mathfrak{N}$, there exists a real number $c_H$ such that $H$ consists of the points $a + t$ of $E$ with $(t|e_H) = c_H$. Since the family $(e_H)$ is free, there exists $t \in T$ such that $(t|e_H) = c_H$ for all $H \in \mathfrak{N}$, and the point $a + t$ of $E$ belongs to all the hyperplanes $H \in \mathfrak{N}$. Thus, (iv) implies (iii).