Oriented k-Sub-Rectangle of Euclidean Space

Let $n,k\in\mathbb{N}$ with $k\le n$. Choose strictly increasing indices

$$1\le i_1<\cdots<i_k\le n,$$

choose real numbers $a_r<b_r$ for $r\in\{1,\dots,k\}$, and for each index $j\in\{1,\dots,n\}\setminus\{i_1,\dots,i_k\}$ choose a real number $c_j$. Let

$$R=[a_1,b_1]\times\cdots\times[a_k,b_k]$$

be the associated standard $k$-rectangle. Define the coordinate insertion map $\lambda_R:R\to\mathbb{R}^n$ by sending

$$(t_1,\dots,t_k)$$

to the point $x=(x_1,\dots,x_n)$ whose coordinates satisfy $x_{i_r}=t_r$ for $r\in\{1,\dots,k\}$ and $x_j=c_j$ for every remaining index $j$.

The image

$$S=\lambda_R(R)\subseteq \mathbb{R}^n$$

is called a $k$-sub-rectangle of $\mathbb{R}^n$. An oriented $k$-sub-rectangle of $\mathbb{R}^n$ is a pair $(S,\varepsilon)$ where $S$ is such a $k$-sub-rectangle and $\varepsilon\in\{1,-1\}$.