Definition 2

A chamber of $E$ relative to $\mathfrak{H}$ (or simply a chamber if there is no ambiguity regarding $\mathfrak{H}$) is a facet of $E$ relative to $\mathfrak{H}$ that is not contained in any hyperplane belonging to $\mathfrak{H}$.

Let $U$ be the open set in $E$ consisting of the points that do not belong to any hyperplane of $\mathfrak{H}$; since a hyperplane of $\mathfrak{H}$ must contain any facet that it meets, the chambers are the facets contained in $U$; every chamber is a convex (hence connected) open subset of $E$ by Prop. 3, (i); since the chambers form a partition of $U$, they are exactly the connected components of $U$. Every convex subset $A$ of $U$ is connected, and thus contained in a chamber, which is unique if $A$ is non-empty. It is clear that the chambers are the facets with support $E$, and Prop. 3, (iii) shows that every chamber is the interior of its closure. Finally, let $C$ be a chamber and $A$ a non-empty subset of $C$; formulas (2) and (3) imply that

since $D_H(A) = D_H(a)$ for all $a \in A$.