# Minkowski distance

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 $p$  (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 level set of the distance function where all points are at the unit distance from the center) with various values of $p$ :