# Minkowski distance

The Minkowski distance is a metric on Euclidean space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

## Definition

The Minkowski distance of order p between two points

$P=(x_1,x_2,\ldots,x_n)\text{ and }Q=(y_1,y_2,\ldots,y_n) \in \mathbb{R}^n$

is defined as:

$\left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}$

For $p\geq1$, 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.

Minkowski distance is typically used with p being 1 or 2. The latter is the Euclidean distance, while the former is sometimes known as the Manhattan distance. 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 with various values of p: