Definition:P-Product Metric
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 $\MM_p := \struct {A_{1'} \times A_{2'}, d_p}$ 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:
- $\ds \map {d_p} {x, y} := \sqrt [p] {\size {x_1 - y_1}^p + \size {x_2 - y_2}^p}$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
General Definition
The definition can be extended to the cartesian product of any finite number $n$ of metric spaces.
Let $M_{1'} = \struct {A_{1'}, d_{1'} }, M_{2'} = \struct {A_{2'}, d_{2'} }, \ldots, M_{n'} = \struct {A_{n'}, d_{n'} }$ 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 $M_{1'}, M_{2'}, \ldots, M_{n'}$.
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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:
- $\map {d_1} {x, y} := \map {d_{1'} } {x_1, y_1} + \map {d_{2'} } {x_2, y_2}$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.
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.