Characterization of N-Cube

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

Let $x, y \in \R^n$, were $x = \left({x_1, x_2, \ldots, x_n}\right)$ and $y = \left({y_1, y_2, \ldots, y_n}\right)$.

Fix $R > 0$.

Let:


 * $\displaystyle Q = \left \{ {x,y \in \R^n : \sup_{x,y} \max_i  \left \vert { y_i - x_i } \right \vert  \le R } \right\}$

Then $Q$ is an $n$-cube.

Proof
For ease of presentation, denote:


 * $y - x = r \in \R^n$

and:


 * $y_j - x_j = r_j$

for $j = 1,2, \ldots\, n$.