Proposition 7

Let L be an affine subspace of E and $\Omega$ a non-empty open subset of L.

(i) There exists a point a in $\Omega$ that does not belong to any of the hyperplanes of $\mathfrak{H}$ that do not contain L.

(ii) If L is a hyperplane and $L \notin \mathfrak{H}$, there exists a chamber that meets $\Omega$.

(iii) If L is a hyperplane and $L \in \mathfrak{H}$, there exists a point a in $\Omega$ that does not belong to any hyperplane $H \neq L$ of $\mathfrak{H}$.

Denote by $\mathfrak{N}$ the set of hyperplanes H with $H \in \mathfrak{H}$ and $L \not\subset H$, and by $\mathscr{L}$ the set of hyperplanes of the affine space L of the form $L \cap H$ with $H \in \mathfrak{N}$. It is clear that $\mathscr{L}$ is a locally finite set of hyperplanes in L, and Prop. 6 shows that $\Omega$ meets a chamber $\Gamma$ defined by $\mathscr{L}$ in L. If $a$ is a point of $\Gamma \cap \Omega$, then $a \notin H$ for all $H \in \mathfrak{N}$, hence (i).

Assume now that L is a hyperplane; any hyperplane containing L is then equal to it, so we may distinguish two cases:

a) $L \notin \mathfrak{H}$: then $\mathfrak{N} = \mathfrak{H}$, and we have $a \notin H$ for all $H \in \mathfrak{H}$; thus $a$ belongs to a chamber defined by $\mathfrak{H}$ in E, hence (ii).

b) $L \in \mathfrak{H}$: then $\mathfrak{N} = \mathfrak{H} - \{L\}$, hence (iii).