Definition 3

Let C be a chamber of E. A face of C is a facet contained in the closure of C whose support is a hyperplane. A wall of C is a hyperplane that is the support of a face of C.

Every wall of C belongs to $\mathfrak{H}$. Prop. 4 shows that a hyperplane $L \in \mathfrak{H}$ is a wall of C if and only if $C \neq D_{\mathfrak{H} - \{L\}}(C)$. Moreover, every wall of C is the support of a single face of C.