Definition:N-Cube (Euclidean Space)

Definition
Let $R, c \in \R$ be real numbers with $R > 0$.

Let $\left({\R^n, d}\right)$ be a Euclidean $n$-Space equipped with the usual metric $d$.

An $n$-cube is a subset of $\left({\R^n, d}\right)$ defined as the cartesian product of closed real intervals of the form:


 * $\displaystyle \prod_{i \mathop = 1}^n \left[{c - R \,.\,.\, c + R}\right]_i$

where $\left[{c - R \,.\,.\, c + R}\right]_i$ is an interval in the $i$th factor of $\R^n$.

The $n$-cube can be concisely expressed as:


 * $\left[{c - R \,.\,.\, c + R}\right]^n$

in contexts where the indices of the product are unimportant.

Also known as
An $n$-cube is sometimes simply called a cube.