= Solution
For the chosen simple roots $\Delta$, the open <fundamental chamber of a root system> is
$$
\boxed{C=\{\lambda\in E:(\lambda,\alpha)>0\text{ for every }\alpha\in\Delta\}.}
$$
Equivalently $\lambda(h_\alpha)>0$ for every simple coroot. Since all positive roots are nonnegative combinations of simple roots, requiring the inequality for all positive roots gives the same chamber. It is a connected component of the complement of the root reflection hyperplanes; the <Weyl group> permutes those chambers simply transitively.
The <Closed dominant Weyl chamber> is obtained by replacing the inequalities by $\ge0$. It is the cone generated by the <fundamental weights>. When the fundamental Weyl chamber is taken to include its walls, this closed convention is intended; both conventions determine the same positive system and boundary rays.
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