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:

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

The Minkowski distance is a metric as a result of the Minkowski inequality.

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:

$\lim_{p\to\infty}{\left(\sum_{i=1}^n |x_i-y_i|^p\right)^\frac{1}{p}} = \max_{i=1}^n |x_i-y_i|. \,$

The following figure shows unit circles with various values of p: