Definition:General Euclidean Metric
Definition
Let $\R^n$ be an $n$-dimensional real vector space.
The general Euclidean metrics are defined on $\R^n$ for $p \in \R_{\ge 1}$ as:
- $\ds \map {d_p} {x, y} := \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{\frac 1 p}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
That is, they are the $p$-product metrics on $\R^n$.
Special Cases
Some special cases of the general Euclidean metric are:
Standard Discrete Metric
The (standard) discrete metric on $\R^2$ is defined as:
- $\map {d_0} {x, y} := \begin {cases}
0 & : x = y \\ 1 & : \exists i \in \set {1, 2}: x_i \ne y_i \end {cases}$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
Taxicab Metric
The taxicab metric on $\R^n$ is defined as:
- $\ds \map {d_1} {x, y} := \sum_{i \mathop = 1}^n \size {x_i - y_i}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
Euclidean Metric
The Euclidean metric on $\R^n$ is defined as:
- $\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
Maximum Metric
The Chebyshev distance on $\R^n$ is defined as:
- $\ds \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
Also see
Note
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Note that while $d_1, d_2, \ldots, d_\infty$ are all topologically equivalent, this is not the case with $d_0$.