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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.

## Definition

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

${\displaystyle X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and ))Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n))$
is defined as:
${\displaystyle D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p)).}$

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

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

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

Similarly, for ${\displaystyle p}$ reaching negative infinity, we have:

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

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

The following figure shows unit circles (the level set of the distance function where all points are at the unit distance from the center) with various values of ${\displaystyle p}$: