Remark

Assume that m is finite and that $\theta = \pi / m$. If $n \in \mathbf{Z}$, let $C_n$ be the union of the open half-lines $\Delta_1$ starting at 0 such that

The $C_n$ for $-m \leq n < m$ are connected open subsets forming a partition of $E - \bigcup_n D_n$ (where $D_n = \rho_{n \theta}(D)$). These are therefore the chambers determined in E by the system of m lines $D_n$ ($1 \leq n \leq m$). We have $C_{2k} = \rho_{2k \theta}(C)$ and $C_{2k-1} = \rho_{2k \theta s}(C)$. Moreover, $C_n = C$ if and only if $n \in 2m \mathbf{Z}$. Consequently, the group W permutes the chambers $C_n$ simply-transitively.

We show finally that, if $w \in W$ is such that the chambers C and $w(C)$ are on opposite sides of the line D, then $l(sw) = l(w) - 1$ (the lengths being taken with respect to $S = \{s, s'\}$). Indeed, the assumption implies that $w(C) = C_n$ with $-m \leq n < 0$. If $n = -2k$, we have $w = (ss')^k$ and $sw = s'(ss')^{k-1}$, so $l(w) = 2k$ and $l(sw) = 2k - 1$ (Chap. IV, § 1, no. 2, Remark). If $n = -2k + 1$, we have

hence $l(w) = 2k - 1$ and $l(sw) = 2k - 2$. Q.E.D.