Corollary 1

Let $P$ be a subset of $R$. The following conditions are equivalent:

(i) there exists a chamber $C$ such that $P = R_+(C)$; (ii) $P$ is closed and $\{P, -P\}$ is a partition of $R$.

The chamber $C$ such that $P = R_+(C)$ is then unique.

If $P = R_+(C)$, $C$ is the set of $x^* \in V^*$ such that $\langle x^*, x \rangle > 0$ for all $x \in P$, hence the uniqueness of $C$.