Relation between P-Product Metric and Chebyshev Distance on Real Vector Space

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For $n \in \N$, let $\R^n$ be a Euclidean space.

Let $p \in \R_{\ge 1}$.

Let $d_p$ be the $p$-product metric on $\R^n$.

Let $d_\infty$ be the Chebyshev distance on $\R^n$.


$\forall x, y \in \R^n: \map {d_\infty} {x, y} \le \map {d_p} {x, y} \le n^{1/p} \map {d_\infty} {x, y}$


By definition of the Chebyshev distance on $\R^n$, we have:

$\ds \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \size {x_i - y_i}$

where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.

Let $j$ be chosen so that:

$\ds \size {x_j - y_j} = \max_{i \mathop = 1}^n \size {x_i - y_i}$


\(\ds \map {d_\infty} {x, y}\) \(=\) \(\ds \paren {\size {x_j - y_j}^p}^{1/p}\)
\(\ds \) \(\le\) \(\ds \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{1/p}\) Power of Maximum is not Greater than Sum of Powers
\(\ds \) \(=\) \(\ds \map {d_p} {x, y}\)
\(\ds \) \(\le\) \(\ds \paren {n \size {x_j - y_j}^p}^{1/p}\) Sum of $r$ Powers is not Greater than $r$ times Power of Maximum
\(\ds \) \(=\) \(\ds n^{1/p} \size {d_\infty} {x, y}\)