Definition:N-Cube (Euclidean Space)

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

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

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


 * $\ds \prod_{i \mathop = 1}^n \closedint {c - R} {c + R}_i$

where $\closedint {c - R} {c + R}_i$ is an interval in the $i$th factor of $\R^n$.

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


 * $\closedint {c - R} {c + R}^n$

in contexts where the indices of the product are unimportant.

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