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:

Also see

 * $p$-Product Metric on Real Vector Space is Metric


 * $p$-Product Metrics on Real Vector Space are Topologically Equivalent


 * Definition:Standard Discrete Metric

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