Proposition 22

*Let P be a closed subset of R such that $P \cap (-P) = \varnothing$. Then there exists a chamber C of R such that $P \subseteq R_+(C)$.

1) In view of the Cor. of Th. 1, no. 3, the assumptions $\alpha \in P, \beta \in P, (\alpha|\beta) < 0$ imply that $\alpha + \beta \in P$.

2) We show that no sum $\alpha_1 + \cdots + \alpha_q$ ($q \geqslant 1$) of elements of P is zero. We proceed by induction on $q$. The assertion being clear for $q = 1$, assume that $q \geqslant 2$. If $\alpha_1 + \cdots + \alpha_q = 0$, then

so $(-\alpha_1|\alpha_2 + \cdots + \alpha_q) > 0$, hence there exists $j \in \{2, q\}$ such that $(\alpha_1|\alpha_j) < 0$. By part 1) of the proof, $\alpha_1 + \alpha_j \in P$, and the relation $(\alpha_1 + \alpha_j) + \sum_{i \neq 1, j} \alpha_i = 0$ contradicts the induction hypothesis.

3) We show that there exists a non-zero element $\gamma$ in $V$ such that $(\gamma|\alpha) \geqslant 0$ for all $\alpha \in P$. If not, the result of 1) would show that an infinite sequence $\alpha_1, \alpha_2, \ldots$ of elements of $P$ could be found such that

for all $i$; there would exist two distinct integers $i, j$ such that $\beta_i = \beta_j$, which would contradict the result of 2).

4) To prove the proposition, it is enough (Cor. 2 of Prop. 20) to show that there exists a basis $(\alpha_k)_{1 \leq k \leq l}$ of $V$ such that, for the lexicographic order defined by this basis, every element of $P$ is $> 0$. We proceed by induction on $l = \dim V$, and assume that the proposition is established for all dimensions $< l$. Let $\gamma \in V$ be such that $\gamma \neq 0$ and $(\gamma|\alpha) \geqslant 0$ for all $\alpha \in P$ (cf. 3)). Let $L$ be the hyperplane orthogonal to $\gamma$, and $V'$ the subspace of $L$ generated by $R \cap L$. Then $R \cap L$ is a root system in $V'$ and $P \cap L$ is closed in $R \cap L$. By the induction hypothesis, there exists a basis $(\beta_1, \ldots, \beta_{l'})$ of $V'$ such that the elements of $P \cap L$ are $> 0$ for the lexicographic order defined by this basis. Then any basis of $V$ whose first $l' + 1$ elements are $\gamma, \beta_1, \ldots, \beta_{l'}$ and whose remaining elements are in $L$ has the required property.