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
$X = (x_1,x_2,\ldots,x_n) \text Y = (y_1,y_2,\ldots,y_n) \in \R^n$
is defined as:
$D\left(X,Y\right) = \left(\sum_^n , x_i-y_i, ^p\right)^.$

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 $2^ > 2,$ 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:
$\lim_ = \max_^n , x_i-y_i, .$

Similarly, for $p$ reaching negative infinity, we have:
$\lim_ = \min_^n , x_i-y_i, .$

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 $p$:

See also

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* {{annotated link, p-norm, $p$-norm

External links

Simple IEEE 754 implementation in C++

NPM JavaScript Package/Module

Normed spaces
Metric geometry
Hermann Minkowski
Distance