with*]{} $\pmb{x}_i$. Then those dividers $\pmb{x}_1^{\perp},\ldots,\pmb{x}_m^{\perp}$ carve up $X$ into a number of parts, and again each part is called a region of $X$. One can readily see that those two ways of defining region are equivalent.

Although a region contains lots of vectors, we can always use one vector to indicate the region. Let us present several notions relevant step by step. We first consider a region $R$ which is not contained in any divider $\pmb{b}_k^{\perp}$ ($k = 1,\ldots,m$). A non-trivial vector $\pmb{s}_i$ of some divider $\pmb{b}_i^{\perp}$ is said to be a [*straightforward projection*]{} of $R$ if there exists $\pmb{r} \in R$ so that $\pmb{s}_i = {\mathrm{proj}\big[  \pmb{b}_i^{\perp}  \big]}( \pmb{r} )$ and $\theta ( \pmb{r} - \pmb{s}_i ) + \pmb{s}_i \in R$, $\forall \theta \in (0,1)$. We call a divider $\pmb{b}_i^{\perp}$ of $\operatorname{\mathbb{R}}^n$ a [*separator*]{} of $R$ if the subspace $${\mathrm{span} \hspace{0.4mm} \{ \pmb{s}_i \in \pmb{b}_i^{\perp} : \pmb{s}_i \mbox{ is a straightforward projection of } R \}} = \pmb{b}_i^{\perp}.$$ The [*incidence set*]{} of $R$, which is denoted by $\mathcal{I}_R$, is defined as follows: $$\mathcal{I}_R = \{ i \in [m] : \pmb{b}_i^{\perp} \mbox{ is a separator of } R \}.$$ Furthermore, we define a sign function on the group $\{ 