Let $\mathcal{C}_{\mathbb{L}}$ be a $\pl$-chamber structure.
The collection $$ \mathcal{C}_{S} = \{ \mathcal{D}\cap\mathcal{P}_{S}\mid\DD\in\mathcal{C}_{\LL}, \; \text{Int}(\mathcal{D}\cap \mathcal{P}_{S}) \neq \emptyset\} $$ is a chamber structure on $\ps$.