dimension $\dim(V)-2$) of the convex polyhedral cone $\overline{T}$ is acute, that is, less than or equal to $\pi/2$.

\[prop\_crossing\_wall\] Suppose that the arrangement ${\mathcal{H}}$ satisfies condition \[(A)\]. Let $T,T'\in{\mathscr{T}}$ and $e \in E$ such that $S(T,T')=\{e\}$, the inner product $(\alpha_e,\lambda)$ is nonnegative and the inclusion $T' \subset H^-_{\alpha_e}$ holds. Then $T'\in{\mathscr{L}}_{\lambda}$ implies that $T\in{\mathscr{L}}_\lambda$.

The hypothesis implies that $\overline{T}\cap\overline{T}'
=
\overline{T}\cap H_{\alpha_e}
=
\overline{T}'\cap H_{\alpha_e}$. We denote this intersection by $S$. It is a facet of both $\overline{T}$ and $\overline{T}'$. By condition \[(A)\], we have $T \subset \operatorname{relint}{S}+{{\mathbb{R}}}_{> 0}\alpha_e$ and $T' \subset \operatorname{relint}{S}+{{\mathbb{R}}}_{< 0}\alpha_e$. In particular, if $x\in T$ then there exists $c>0$ such that $x - c \cdot \alpha_e \in T'$. Then we have $(x,\lambda)=(x- c \cdot \alpha_e,\lambda)+c \cdot (\alpha_e,\lambda)\geq 0$. This implies that $T\subset\overline{H_\lambda^+}$, that is, $T\in{\mathscr{L}}_{\lambda}$.

Suppose that the arrangement ${\mathcal{H}}$ satisfies condition \[(A)\]. If $(\lambda,\alpha_e)\geq 0$ for every $e \in E$ and if there exists $B\in{\mathscr{T}}$ such that $B\subset H^+_{\alpha_e}$ for every $e \in E$, then ${\mathscr{T}}\cap {\mathscr{L}}_{\lambda}$ is a lower order ideal in ${\mathscr{T}}_B$. More generally, if $C\in{\mathscr{L}}$ and $E(C)=\{e\in E :  C\subset H_{\alpha_e}\}$, if $(\lambda,\alpha_f)\geq 0$ for every $f\in E(C)$ and if there exists $B\in{\mathscr{T}}\cap{\mathscr{L}}_{\geq C}$ such that $B\subset H^+_{\alpha_f}$ for every $f\in E(C)$, 