= Solution
Choose a positive system $R^+$ and its <simple roots> $\Delta=\{\alpha_1,\ldots,\alpha_\ell\}$. The corresponding fundamental <Weyl chamber> is the open cone
$$
\boxed{\mathcal W=\{\lambda\in E:\langle\lambda,\alpha_i^\vee\rangle>0\text{ for all }i\}.}
$$
Its <Closed dominant Weyl chamber> replaces all strict inequalities by weak inequalities. The facts used here are that every <positive root> is a nonnegative integral combination of the <simple roots>, and that the <Weyl chambers> are the connected components left after removing the root-orthogonal hyperplanes. Hence the displayed inequalities describe one such component. The <Weyl group> permutes these components simply transitively, and every orbit in $E$ meets the <Closed dominant Weyl chamber> in exactly one point. A point on a wall may have a nontrivial stabilizer, which is why uniqueness of the chamber point does not always mean uniqueness of the group element carrying it there.
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