Proposition 11

Assume that $W$ is essential. Let $a$ be a special point for $W$. The chambers relative to $W_a$ are the open simplicial cones with vertex $a$. For every chamber $C'$ relative to $W_a$, there exists a unique chamber $C$ relative to $W$ contained in $C'$ and such that $a \in \overline{C}$. The union of the $w'(\overline{C})$ for $w' \in W_a$ is a closed neighbourhood of $a$ in $E$. Every wall of $C'$ is a wall of $C$. If $W$ is infinite and irreducible, the walls of $C$ are the walls of $C'$ together with an affine hyperplane not parallel to the walls of $C'$.

Let $\mathfrak{H}'$ be the set of $H \in \mathfrak{H}$ containing $a$. The group $W_a$ is generated by the $s_H$ for $H \in \mathfrak{H}'$ (no. 3, Prop. 2). The chambers relative to $W_a$ are the open simplicial cones with vertex $a$ (no. 9, Prop. 7). Let $C'$ be such a chamber and let $U$ be a non-empty open ball with centre $a$ not meeting any element of $\mathfrak{H} - \mathfrak{H}'$. Since $a \in \overline{C'}$, there exists a point $b$ in $U \cap C'$. Then $b \notin H$ for all $H \in \mathfrak{H}$, so $b$ belongs to a chamber $C$ relative to $\mathfrak{H}$. Since $\mathfrak{H}' \subset \mathfrak{H}$, we have $C \subset C'$. The set $U \cap C'$ does not meet any $H \in \mathfrak{H}$ and is convex, so $U \cap C' \subset C$; thus $a \in \overline{C}$. Conversely, let $C_1$ be a chamber relative to $W$ contained in $C'$ and such that $a \in \overline{C}_1$; then $C_1$ meets $U$ and $U \cap C_1 \subset U \cap C' = U \cap C$; the chambers $C$ and $C_1$, having a point in common, must coincide. For any $w' \in W_a$, we have $w'(U) = U$, so

since the union of the $w'(C)$ for $w' \in W_a$ is dense in $E$, the union of the $U \cap w'(C) = U \cap w'(C')$ is dense in $U$, and the union of the $w'(\overline{C})$ for $w' \in W_a$ thus contains $U$. Finally, if $H$ is a wall of $C'$, there exist a point $c \in U \cap H$ and an open neighbourhood $V \subset U$ of $c$ such that $V \cap C'$ is the intersection of $V$ and the open half-space bounded by $H$ containing $C'$; since $V \cap C' = V \cap U \cap C' = V \cap U \cap C = V \cap C$, it follows that $H$ is a wall of $C$. If $W$ is infinite and irreducible, $C$ is an open simplex (no. 9, Prop. 8) and so has one more wall than the open simplicial cone $C'$.