# Minkowski distance

The **Minkowski distance** is a metric in a normed vector 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

is defined as:

For , the Minkowski distance is a metric as a result of the Minkowski inequality. When , the distance between (0,0) and (1,1) is , but the point (0,1) is at a distance 1 from both of these points. Since this violates the triangle inequality, for it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of . 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:

Similarly, for *p* reaching negative infinity, we have:

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 set of all points that are at the unit distance from the centre) with various values of *p*: