Definition:P-Product Metric/General Definition

Definition

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

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

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

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

$\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^p}^{\frac 1 p}$

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

The metric space $\MM_p := \struct {\AA, d_p}$ 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:

$\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p}$

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

Special Cases

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

Taxicab Metric

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

$\ds \map {d_1} {x, y} := \sum_{i \mathop = 1}^n \map {d_{i'} } {x_i, y_i}$

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

Euclidean Metric

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

$\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$

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

Chebyshev Distance

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

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

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

Standard Discrete Metric

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

$\map {d_0} {x, y} := \begin {cases}

0 & : x = y \\ 1 & : \exists i \in \set {1, 2}: x_i \ne y_i \end {cases}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.

Also see

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

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.