Remark 2

Let $F$ be a facet whose support is a hyperplane $L$; we shall show that $F$ is a face of two chambers. Let $\mathfrak{N}$ be the set of hyperplanes $H \neq L$ belonging to $\mathfrak{H}$; put $A = D_{\mathfrak{N}}(F)$ and denote by $D^+$ and $D^-$ the two open half-spaces bounded by $L$. The set $A$ is open and contains $F \subset L$, and since every point of $L$ is in the closure of $D^+$ and $D^-$, the sets $C^+ = A \cap D^+$ and $C^- = A \cap D^-$ are non-empty; these are chambers. Moreover, the hyperplane $L$ meets $D_{\mathfrak{N}}(F) = D_{\mathfrak{N}}(C^+)$; Prop. 4 shows that $L$ is a wall of $C^+$ and that

F, which meets $L \cap D_{\gamma_l}(F)$, is a face of $C^+$; similarly, F is a face of $C^-$. Finally, let C be a chamber of which F is a face, and suppose for example that $D^+ = D_L(C)$; by Prop. 4, the set $D_{\gamma}(C)$ meets F, and hence is equal to $D_{\gamma}(F)$, and we have