Characterization of N-Cube

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Theorem

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

Let $x, y \in \R^n$, where $x = \left({x_1, x_2, \ldots, x_n}\right)$ 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 $j = 1,2, \ldots\, n$.

\(\displaystyle Q\) \(=\) \(\displaystyle \left \{ {r : \sup_r \max_i \left \vert { r_i } \right \vert \le R } \right\}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left \{ {r : \sup_r \max \left({ \vert r_1 \vert, \vert r_2 \vert, \ldots, \vert r_n \vert}\right) \le R } \right\}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left \{ { r :\sup_\ell r_1 \in \left [{-\ell \,.\,.\, \ell} \right], r_2 \in \left [{-\ell \,.\,.\, \ell} \right], \ldots, r_n \in \left [{-\ell \,.\,.\, \ell} \right], \ell \le R }\right\}\) $\quad$ where $\ell = \max \left({ \vert r_1 \vert, \vert r_2 \vert, \ldots, \vert r_n \vert}\right)$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \prod_{i \mathop = 1}^n \left [{-R \,.\,.\, R} \right]_i\) $\quad$ $\quad$

$\blacksquare$


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