= Solution
Let $C(\Delta)$ be the <fundamental chamber of a root system> determined by $\Delta$. Every positive root is a nonnegative linear combination of the simple roots, so every point in the interior of $C(\Delta)$ pairs strictly positively with every positive root. In particular, the interior meets none of the reflecting hyperplanes belonging to the subsystem $\Phi_0$ of roots of maximal length.
The connected set $C(\Delta)$ therefore lies in one chamber of the <long-root subsystem>. Let $\Delta_0$ be the unique <fundamental system of a root system> defining that chamber. Taking closures gives $C(\Delta)\subseteq C(\Delta_0)$. Uniqueness follows because the interiors of two distinct chambers are disjoint. Thus \b[there is a unique such $\Delta_0$].
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