Proposition 3

Let C be a chamber and let $i, j \in I$ with $i \neq j$. Put $s_i = s_i(C), H_i = H_i(C), e_i = e_i(C)$ and define $s_j, H_j$ and $e_j$ similarly.

(i) If $H_i$ and $H_j$ are parallel, then $m_{ij} = \infty$ and $e_i = -e_j$.

(ii) If $H_i$ and $H_j$ are not parallel, then $m_{ij}$ is finite and

(iii) We have $(e_i|e_j) \leq 0$.

If $H_i$ and $H_j$ are parallel, $s_i s_j$ is a translation (§ 2, no. 4, Prop. 5), so $m_{ij} = \infty$. Moreover, either $e_i = e_j$ or $e_i = -e_j$. Now, there exists a point $a$ (resp. $a'$) in the closure of C that belongs to $H_i$ (resp. $H_j$) but not to $H_j$ (resp. $H_i$). Then $(a' - a|e_i) > 0$ and $(a - a'|e_j) > 0$, which excludes the case $e_i = e_j$ and proves (i).

Assume now that $H_i$ and $H_j$ are not parallel. Choose an origin $a \in H_i \cap H_j$ and identify T with E by the bijection $t \mapsto a + t$. Let V be the plane orthogonal to $H_i \cap H_j$ and passing through $a$. Put $\Gamma = V \cap D_{H_i}(C) \cap D_{H_j}(C)$ (where $D_H(C)$ denotes the open half-space bounded by H and containing C ($§ 1$, no. 1)) and let D (resp. $D'$) be the half-line in V contained in $H_i \cap V$ (resp. $H_j \cap V$) and in the closure of $\Gamma$. For a suitable orientation of V, the set $\Gamma$ is the union of the open half-lines $\Delta$ in V such that

Let $W'$ be the subgroup of W generated by $s_i$ and $s_j$. For any $w \in W'$, the hyperplanes $w(H_i)$ and $w(H_j)$ belong to $\mathscr{H}$, contain $H_i \cap H_j$ and do not meet C. It follows that they do not meet $\Gamma$ ($§ 1$, no. 5, Prop. 10). The Cor. of Prop. 7 of $§ 2$, no. 5 thus implies (ii).

Finally, assertion (iii) follows immediately from (i) and (ii), since $m_{ij} \geq 2$ for $i \neq j$.