Definition:P-Product Metric

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Definition

Let $M_{1'} = \struct {A_{1'}, d_{1'} }$ and $M_{2'} = \struct {A_{2'}, d_{2'} }$ be metric spaces.

Let $A_{1'} \times A_{2'}$ be the cartesian product of $A_{1'}$ and $A_{2'}$.

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


The $p$-product metric on $A_{1'} \times A_{2'}$ is defined as:

$\map {d_p} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^p + \paren {\map {d_{2'} } {x_2, y_2} }^p}^{1/p}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.


The metric space $\mathcal M_p := \struct {A_{1'} \times A_{2'}, d_p}$ is the $p$-product (space) of $M_{1'}$ and $M_{2'}$.


Real Number Plane

This metric is often found in the context of a real number plane $\R^2$:

The $p$-product metric on $\R^2$ is defined as:

$\displaystyle d_p \left({x, y}\right) := \sqrt [p] {\left\vert{x_1 - y_1}\right\vert^p + \left\vert{x_2 - y_2}\right\vert^p}$

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


General Definition

The definition can be extended to the cartesian product of any finite number $n$ of metric spaces.

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 $M_{1'}, M_{2'}, \ldots, M_{n'}$.



Special Cases

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

Taxicab Metric

The taxicab metric on $A_{1'} \times A_{2'}$ is defined as:

$d_1 \left({x, y}\right) := d_{1'} \left({x_1, y_1}\right) + d_{2'} \left({x_2, y_2}\right)$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in A_{1'} \times A_{2'}$.


Euclidean Metric

The Euclidean metric on $A_{1'} \times A_{2'}$ is defined as:

$\map {d_2} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^2 + \paren {\map {d_{2'} } {x_2, y_2} }^2}^{1/2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.


Chebyshev Distance

The Chebyshev distance on $A_1 \times A_2$ is defined as:

$\map {d_\infty} {x, y} := \max \set {\map {d_1} {x_1, y_1}, \map {d_2} {x_2, y_2} }$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_1 \times A_2$.


Also known as

The $p$-product metric is also referred to as the $p$-norm, but that term is also used for a slightly more general concept.


Also see

  • Results about $p$-product metrics can be found here.


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.