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

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:

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

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

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

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