H^+_\alpha & = \{x\in V : (\alpha,x)> 0\}, &
H^-_\alpha & = \{x\in V : (\alpha,x)< 0\}.\end{aligned}$$ We also denote by $s_\alpha$ the (orthogonal) reflection across the hyperplane $H_\alpha$.

Let $(\alpha_e)_{e \in E}$ be a finite family of nonzero vectors in $V$. The corresponding *(central) hyperplane arrangement* is the family of hyperplanes ${\mathcal{H}}= (H_{\alpha_e})_{e \in E}$. Let $V_{0}$ be the intersection of all the hyperplanes, that is, $V_0=\bigcap_{e \in E} H_{\alpha_e}$. We say that the arrangement ${\mathcal{H}}$ is *essential* if $V_0 = \{0\}$, which means that the family $(\alpha_e)_{e \in E}$ spans $V$.

We will mostly be interested in hyperplane arrangements that are essential and have no repeated hyperplanes, that is, where $\alpha_e\not\in{{\mathbb{R}}}\alpha_f$ for distinct $e, f \in E$. However, some of the arrangements arising in the inductive formula of Proposition \[prop\_induction\_formula\_for\_psi\] might not satisfy these conditions, so we do not want to impose this restriction earlier than necessary.

Consider the map $s:V \longrightarrow \{+,-,0\}^E$ sending $x\in V$ to the family $(\operatorname{sign}((\alpha_e,x)))_{e \in E}$, where $\operatorname{sign}:{{\mathbb{R}}}\longrightarrow \{+,-,0\}$ is the map sending positive numbers to $+$, negative numbers to $-$ and zero to $0$.

The image of the map $s:V \longrightarrow \{+,-,0\}^E$ is the set of covectors of an oriented matroid 