Category:P-Product Metrics

This category contains results about $p$-product metrics.
Definitions specific to this category can be found in Definitions/P-Product Metrics.

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 $M_{2'}$.