for each member $\pmb{b}_i$ in the group, there is one unique subspace of dimension $n-1$, which is the orthogonal complement, denoted by $\pmb{b}_i^{ \perp }$, of the vector $\pmb{b}_i$ and called the [*divider of $\operatorname{\mathbb{R}}^n$ associated with $\pmb{b}_i$*]{}. Then the whole space $\operatorname{\mathbb{R}}^n$ is divided into 3 parts by $\pmb{b}_i^{ \perp }$:

1.  those vectors each of which has a positive inner product with $\pmb{b}_i$, so we denote the part by $\pmb{b}_i^+$;

2.  those vectors each of which has a negative inner product with $\pmb{b}_i$, so we denote the part by $\pmb{b}_i^-$;

3.  those vectors each of which is orthogonal to $\pmb{b}_i$, so we denote the part by $\pmb{b}_i^{ \perp }$.

By using all dividers $\pmb{b}_i^{\perp},\ldots,\pmb{b}_m^{\perp}$ of $\operatorname{\mathbb{R}}^n$, the whole space can be divided into many parts of two classes: those each of which is comprised of vectors not orthogonal to any vector in the group $\{ \pmb{b}_i : i \in [m] \}$, and those each of which is contained in some divider. In order to investigate those parts contained in a divider $\pmb{b}_i^{ \perp }$, we focus on dividers of the subspace $\pmb{b}_i^{ \perp }$ associated with vectors $\left\{ {\mathrm{proj}\big[  \pmb{b}_i^{ \perp }  \big]}( \pmb{b}_k ) : k \in 