Definition:General Euclidean Metric

Definition
Let $\R^n$ be an $n$-dimensional real vector space.

Let $x = \left({x_1, x_2, \ldots, x_n}\right) \in \R^n$ and $y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.

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

The general Euclidean metrics are defined for $p \in \R_{\ge 1}$ as:


 * $\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left\vert{x_i - y_i}\right\vert^p}\right)^{\frac 1 p}$

That is, they are the $p$-product metrics on $\R^n$.

Special Cases
Some special cases of the general Euclidean metric are:

Also see

 * General Euclidean Metric is Metric


 * General Euclidean Metrics are Topologically Equivalent


 * Definition:Discrete Metric

Note that $d_2 \left({x, y}\right)$ is the usual Euclidean metric:
 * $\displaystyle d \left({x, y}\right) = \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{\frac 1 2}$

on $\R^n$.

Note
To complete the family, we could also define $d_0$ as the standard discrete metric on $\R^n$.

However, while $d_1, d_2, \ldots, d_\infty$ are all topologically equivalent, this is not the case with $d_0$.