Characterization of N-Cube

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

Let $x, y \in \R^n$, where:
 * $x = \tuple {x_1, x_2, \ldots, x_n}$
 * $y = \tuple {y_1, y_2, \ldots, y_n}$

Let $R > 0$ be fixed.

Let:


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

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