The **Minkowski distance** or **Minkowski metric** is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after the German mathematician Hermann Minkowski.

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Definition

The Minkowski distance of order $p$ (where $p$ is an integer) between two points

$X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and ))Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n))$

is defined as:
$D\left(X,Y\right)=\left(\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}\right)^{\frac {1}{p)).$

For $p\geq 1,$ the Minkowski distance is a metric as a result of the Minkowski inequality. When $p<1,$ the distance between $(0,0)$ and $(1,1)$ is $2^{1/p}>2,$ but the point $(0,1)$ is at a distance $1$ from both of these points. Since this violates the triangle inequality, for $p<1$ it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of $1/p.$ The resulting metric is also an F-norm.

Minkowski distance is typically used with $p$ being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of $p$ reaching infinity, we obtain the Chebyshev distance:

$\lim _{p\to \infty }{\left(\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}\right)^{\frac {1}{p))}=\max _{i=1}^{n}|x_{i}-y_{i}|.$

Similarly, for $p$ reaching negative infinity, we have:

$\lim _{p\to -\infty }{\left(\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}\right)^{\frac {1}{p))}=\min _{i=1}^{n}|x_{i}-y_{i}|.$

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between $P$ and $Q.$

The following figure shows unit circles (the set of all points that are at the unit distance from the center) with various values of $p$: