Definition:P-Product Metric/Real Number Plane

Definition

Let $\R^2$ be the real number plane.

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

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

Also known as

The $p$-product metric is sometimes also referred to as the $r$-product metric by sources which use $r$ for the general power.

Some sources refer to this metric as the general (or generalized) Euclidean metric, and the space $\struct {\R^2, d_p}$ as the general (or generalized) Euclidean plane

Also see

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