Definition:Generalized Euclidean Metric

Definition

$\newcommand{\dist} [2] {\left| {{#1} - {#2}} \right|}$ 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 $y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.

The generalized Euclidean metrics are defined as follows:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle d_1 \left({x, y}\right)\)	\(=\)	\(\displaystyle \sum_{i=1}^n \dist {x_i} {y_i}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle d_r \left({x, y}\right)\)	\(=\)	\(\displaystyle \left({\sum_{i=1}^n \dist {x_i} {y_i}^r}\right)^{\frac 1 r}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	$r = 2, 3, \ldots$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle d_\infty \left({x, y}\right)\)	\(=\)	\(\displaystyle \max_{i=1}^n \left\{ {\dist {x_i} {y_i} }\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

The Generalized Euclidean Metric is a Metric.

The Generalized Euclidean Metrics are Topologically Equivalent.

Note that $d_2 \left({x, y}\right)$ is the usual Euclidean metric:

$\displaystyle d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{\frac 1 2}$

on $\R^n$.

Relationship with Product Space Metrics

It can be seen that this is a special case of a product space.

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 $d_0$.

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Examples $2.2.3, \ 2.2.18$