Proposition 12

Assume that $W$ is essential. Let $a$ be a special point, let $L(a)$ be the set of its transforms under the group of translations $L$, and let $C$ be a chamber. Then $\overline{C}$ meets $L(a)$ in a unique point; this point is an extremal point of $\overline{C}$.

There exists a chamber $C_1$ such that $a$ is an extremal point of $\overline{C}_1$ (Cor. of Prop. 11). Every chamber is of the form $C = t w'(C_1)$ with $w' \in W_a$ and $t \in L$ since $W = W_a . L$; thus $t w'(a) = t(a) \in L(a)$ is an extremal point of $\overline{C}$. On the other hand, $\overline{C}$ cannot contain two distinct points of $L(a)$ since $\overline{C}$ is a fundamental domain for $W$ (no. 3, Th. 2).