Weyl chamber

Remarks

Each chamber is an open convex cone: for any $\gamma$ in the chamber $C$, it equals $\set{x \in E : (x, \alpha) (\gamma, \alpha) > 0 \text{ for all } \alpha \in \Phi}$. Example: for $A_1 = \set{\pm \alpha}$ in $E = \R$, the only hyperplane is $H_\alpha = \set{0}$, and the chambers are $(0, \infty)$ and $(-\infty, 0)$. Non-example: the closed half-line $[0, \infty)$ is not a chamber, since it meets $H_\alpha$. For a base $\Delta$ of simple roots, the set $\set{x \in E : (x, \alpha) > 0 \text{ for all } \alpha \in \Delta}$ is a chamber, the fundamental chamber of $\Delta$; this gives a bijection between bases and chambers. The Weyl group acts simply transitively on the set of chambers.