Characterization of N-Cube

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

\(\displaystyle Q\) \(=\) \(\displaystyle \set {r: \map {\sup_r} {\max_i \size {r_i} \le R} }\)
\(\displaystyle \) \(=\) \(\displaystyle \set {r: \map {\sup_r} {\map \max {\size {r_1}, \size {r_2}, \ldots, \size {r_n} } \le R} }\)
\(\displaystyle \) \(=\) \(\displaystyle \set {r: \map {\sup_\ell} {r_1 \in \closedint {-\ell} \ell, r_2 \in \closedint {-\ell} \ell, \ldots, r_n \in \closedint {-\ell} \ell, \ell \le R} }\) where $\ell = \map \max {\size {r_1}, \size {r_2}, \ldots, \size {r_n} }$
\(\displaystyle \) \(=\) \(\displaystyle \prod_{i \mathop = 1}^n \closedint {-R} R_i\)

$\blacksquare$


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