$H^+_\lambda$. In particular, there exists a chamber $B\in{\mathscr{T}}$ such that $x\in B$, and $B$ is included in $H^+_{\varepsilon_e\alpha_e}$ for every $e\in F$ and in $H^+_\lambda$. Now, if $e\in E -  F$, we choose $\varepsilon_e\in\{\pm 1\}$ such that $B\subset H^+_{\varepsilon_e\alpha_e}$. As $(\lambda,\alpha_e)=0$, we clearly have $(\lambda,\varepsilon_e\alpha_e)\geq 0$.

Coxeter arrangements {#section_Coxeter_arrangements}
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Let $(W,S)$ be a Coxeter system, that is, $W$ is the group generated by the set $S$ and the relations between the generators are of the form $(st)^{m_{s,t}} = 1$ where $m_{s,s} = 1$ and $m_{s,t} \geq 2$ for $s \neq t$; see [@BB Section 1.1]. The corresponding Coxeter graph has vertex set $S$, and two generators $s$ and $t$ are connected with an edge if $m_{s,t} \geq 3$. If $m_{s,t} \geq 4$ it is customary to label the edge by the integer $m_{s,t}$.

There are three natural partial orders on the elements of the Coxeter group $W$. First the [*strong Bruhat order*]{} is defined by the following cover relation: $z {\prec}w$ if there exists $s\in S$ and $u\in W$ such that $(usu^{-1}) z = w$ and $\ell(z)+1 = \ell(w)$ where $\ell$ is the length function on $W$; see for example [@BB Definition 2.1.1]. Next, we have the [*right*]{} (respectively 