Corollary

Assume that $W$ is essential.

(i) If $a \in E$ is special, there exists a chamber $C$ such that $a$ is an extremal point of $\overline{C}$.

(ii) If $C$ is a chamber, there exists an extremal point of $\overline{C}$ that is special.

The first assertion follows from Prop. 11. The second follows from the first and the fact that $W$ acts transitively on the set of chambers.

However, an extremal point of $\overline{C}$ is not necessarily a special point for $W$ (cf. Chap. VI, Plate X, Systems $B_2$ and $G_2$).