Exercise 1, § 7

Let $\omega \in P_{++}$; denote by $S(\omega )$ the set of weights of $E(\omega )$, in other words the smallest R-saturated subset of P containing $\omega$ (Prop. 5). If $\lambda \in S(\omega )$, we have $\lambda \equiv \omega$ (mod. Q). Conversely, let $\lambda \in P$ be such that $\lambda \equiv \omega$ (mod. Q); prove the equivalence of the following properties: (i) $\lambda \in S(\omega )$;

(ii) $\omega -w\lambda \in Q_+$ for all $w\in W$;

(iii) $\lambda$ belongs to the convex hull of $W.\omega$ in $\mathfrak{h}^*_{\mathbf{R}}$.

(To prove that (iii) $=\Rightarrow$ (ii), remark that $\omega -w\omega$ is a linear combination of elements of $R_+$ with coefficients $\geq 0$; deduce, by convexity, that $\omega -w\lambda$ has the same property; since $\omega -w\lambda$ belongs to Q, this implies that $\omega -w\lambda \in Q_+$. To prove that (ii) $=\Rightarrow$ (i), choose $w$ such that $w\lambda \in P_{++}$, and apply Cor. 2 of Prop. 3. The implication (i) $=\Rightarrow$ (iii) is immediate.)