\pmb{b}_i : i \in [m] \}$ related to $R$: $${\mathrm{sgn} \hspace{0.2mm} [R] (\pmb{b}_i)} = \left\{ 
\begin{matrix}
1  & \mbox{ if } \pmb{b}_i^{\perp} \mbox{ is a separator of } R & \hspace{-2mm} \mbox{ and } R \subseteq \pmb{b}_i^{+}; \\
-1 & \mbox{ if } \pmb{b}_i^{\perp} \mbox{ is a separator of } R & \hspace{-2mm} \mbox{ and } R \subseteq \pmb{b}_i^{-}; \\
0  & \hspace{2mm} \mbox{ otherwise. } & ~
\end{matrix}
\right.$$ We are now ready to introduce the [*indicator of $R$*]{}, which is $$\mbox{the vector } \sum_{ x\in\mathcal{I}_R } \frac{{\mathrm{sgn} \hspace{0.2mm} [R] (\pmb{b}_x)}}{\|\pmb{b}_x\|} \cdot \pmb{b}_x \mbox{ and denoted by } \pmb{i}_R.$$ It is easy to see that the indicator $\pmb{i}_R$ is contained in $R$.

Note that we assumed that the region $R$ considered is not contained in any divider involved, but it could be the case that there are some of dividers, $\pmb{b}^{\perp}_{ k_1 },\ldots,\pmb{b}^{\perp}_{ k_q }$, say, such that $R \subseteq \cap_{i\in [q]} \pmb{b}^{\perp}_{ k_i }$. Accordingly we should focus on the division of the subspace $X = \cap_{i\in [q]} \pmb{b}^{\perp}_{ k_i }$ with respect dividers $\big\{ \pmb{p}_j^{\perp} : j \in [m] \setminus \{ k_1,\ldots,k_q \} \big\},$ where $\pmb{p}_j = {\mathrm{proj}\big[  X  \big]}( \pmb{b}_j )$. Then we could define those 