Definition:P-Product Metric/General Definition

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Definition

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces.

Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.

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


The $p$-product metric on $\mathcal A$ is defined as:

$\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.


The metric space $\mathcal M_p := \left({\mathcal A, d_p}\right)$ is the $p$-product (space) of $M_{1'}, M_{2'}, \ldots, M_{n'}$.


Real Vector Space

This metric is often found in the context of a real vector space $\R^n$:

The $p$-product metric on $\R^n$ is defined as:

$\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left\vert{x_i - y_i}\right\vert^p}\right)^{\frac 1 p}$

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


Special Cases

Some special cases of the $p$-product metric are:


Taxicab Metric

The taxicab metric on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as:

$\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n d_{i'} \left({x_i, y_i}\right)$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.


Euclidean Metric

The Euclidean metric on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as:

$\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^2}\right)^{\frac 1 2}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.


Chebyshev Distance

The Chebyshev distance on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ is defined as:

$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$

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


Standard Discrete Metric

The discrete metric on $\R^2$ is defined as:

$\displaystyle d_0 \left({x, y}\right) := \begin{cases} 0 & : x = y \\ 1 & : \exists i \in \left\{{1, 2}\right\}: x_i \ne y_i \end{cases}$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.


Also see


Note

Note that while $d_1, d_2, \ldots, d_\infty$ are all topologically equivalent, this is not the case with $d_0$.


Notation

The notation is awkward, because it is necessary to use a indexing subscript for the $n$ metric spaces contributing to the product, and also for the $p$th exponential that defines the metric itself.

Thus the "prime" notation on the $n$ metric spaces.