### 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*:

## See also

## External links

Simple IEEE 754 implementation in C++