📂Geometry

Definition of a General Parallelepiped

Definition

Given $n$ linearly independent vectors $y_{1},\ \cdots,\ y_{n} \in \mathbb{R}^n$, the set $P$ is known as a parallelepiped.

$$P = \left\{ \sum \limits_{j=1}^{n} \lambda_{j} y_{j} \ \ \Big| \quad 0\le \lambda_{j} \le 1 \right\}$$

Description

As defined, it includes the origin as a vertex. Simply put, it is the set of all linear combinations with coefficients up to 1.

For $n=3$, it forms a parallelepiped, and for $n=2$, it forms a parallelogram.

The collection of all points within and on the boundary of the parallelogram (parallelepiped) shown in the figure aligns with the defined $P$.

Also, for $x\in \mathbb{R}^{n}$, $x+P$ becomes the set $P$ translated, and $x+P$ includes $x$ as one of its vertices.

Let’s denote the center of $P$ as $c(P)$. Then, the following holds true.

$$c(P)=\frac{1}{2}\left(y_{1}+\cdots+y_{n} \right)$$

Thinking of the case of a parallelogram in two dimensions, this should be immediately convincing. In the case of translating by $x$, it looks like this.

$$c(x+P)=x+\frac{1}{2}\left( y_{1} + \cdots y_{n} \right)$$