Proposition 6

Let $\lambda \in P$, and $\mathscr{X}$ the smallest R-saturated subset of P containing $\lambda$. Choose a norm $\| \cdot \|$ as in Prop. 5. The following conditions are equivalent:

(i) $\mathscr{X}= W.\lambda$;

(ii) all the elements of $\mathscr{X}$ have the same norm;

(iii) for all $\alpha \in R$, we have $\lambda (H_{\alpha})\in \{0,1,-1\}$.

Every non-empty R-saturated subset of P contains an element $\lambda$ satisfying the above conditions.

Introduce the condition:

(ii$'$) for all $\alpha \in R$ and for every integer $t$ between 0 and $\lambda (H_{\alpha})$,

(i) $=\Rightarrow$ (ii) $=\Rightarrow$ (ii$'$)$:$ This is clear.

(ii$'$) $=\Rightarrow$ (iii): Assume that condition (ii$'$) is satisfied. Let $\alpha \in R$. We have $\|\lambda \|=\|\lambda -\lambda (H_{\alpha})\alpha \|$, so $\|\lambda -t\alpha \|<\|\lambda \|$ for every integer $t$ strictly between 0 and $\lambda (H_{\alpha})$; hence, there can be no such integers, so $|\lambda (H_{\alpha})| \leq 1$.

(iii) $=\Rightarrow$ (i): Assume that condition (iii) is satisfied. Let $w\in W$ and $\alpha \in R$. Then $(w\lambda )(H_{\alpha}) =\lambda (H_{w^{-1}\alpha})\in \{0,1,-1\}$; thus, if $t$ is an integer between 0 and $(w\lambda )(H_{\alpha}),w\lambda -t\alpha$ is equal to $w\lambda$ or $s_{\alpha}(w\lambda )$. This proves that $W.\lambda$ is R-saturated, so $\mathscr{X}= W.\lambda$.

Let $\mathscr{Y}$ be a non-empty R-saturated subset of P. There exists in $\mathscr{Y}$ an element $\lambda$ of minimum norm. It is clear that $\lambda$ satisfies condition (ii$'$), hence the last assertion of the proposition.