point $x$ belongs to the cone $C$. This is a contradiction since $C$ is contained in the half-space $\overline{H_{\lambda}^+}$, so $(\lambda,x) \geq 0$.

The subcomplex of the cell decomposition $\Sigma({\mathscr{L}})$, respectively $\Sigma({\mathscr{L}}_{\geq C_0})$, whose face poset is the lower order ideal ${\mathscr{L}}_\lambda$, respectively ${\mathscr{L}}_{\lambda,\geq C_0}$, is called the *weighted complex* and denoted by $\Sigma({\mathscr{L}}_\lambda)$, respectively $\Sigma({\mathscr{L}}_{\lambda,\geq C_0})$. By Proposition \[prop\_weighted\_complex\_basic\] it is pure of the same dimension as $\Sigma({\mathscr{L}}_\lambda)$, respectively $\Sigma({\mathscr{L}}_{\lambda,\geq C_0})$.

We now introduce a geometric condition on the hyperplane arrangement ${\mathcal{H}}$ that will imply the shellability of the weighted complex.

Denote by [ (A)currentlabel\[(A)\]]{} the following condition on the family $(\alpha_e)_{e \in E}$ (or the corresponding arrangement): For every $T\in{\mathscr{T}}$ and for every $e \in E$ such that $S = \overline{T}\cap
H_{\alpha_e}$ is of dimension $\dim(V)-1$, that is, $S$ is a facet of the convex cone $\overline{T}$, the following inclusions hold: $$\begin{aligned}
T\subseteq\operatorname{relint}(S)+{{\mathbb{R}}}_{> 0}\alpha_e \text{ if } T\subseteq H^+_{\alpha_e}, \\
T\subseteq\operatorname{relint}(S)+{{\mathbb{R}}}_{< 0}\alpha_e \text{ if } T\subseteq H^-_{\alpha_e},\end{aligned}$$ where $\operatorname{relint}{S}$ is the relative interior of the cone $S$, that is, the interior of $S$ in $\operatorname{Span}(S)$.

Geometrically, condition \[(A)\] means that if $T\in{\mathscr{T}}$ then the dihedral angle between any two adjacent facets (facets whose intersection is a face of 