Minkowski distance

Not to be confused with the pseudo-Euclidean metric of the Minkowski space.

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.

Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard

Definition

The Minkowski distance of order $p$ (where $p$ is an integer) between two points

$$X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n}$$

is defined as:

$$D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.$$

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 $2^{1/p}>2,$ 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. 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:

$$\lim _{p\to \infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\max _{i=1}^{n}|x_{i}-y_{i}|.$$

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

$$\lim _{p\to -\infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\min _{i=1}^{n}|x_{i}-y_{i}|.$$

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 (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

• Generalized mean – N-th root of the arithmetic mean of the given numbers raised to the power n
• $L^{p}$ space – Function spaces generalizing finite-dimensional p norm spaces
• Norm (mathematics) – Length in a vector space
• $p$-norm – Function spaces generalizing finite-dimensional p norm spacesPages displaying short descriptions of redirect targets

External links

• Unit Balls for Different p-Norms in 2D and 3D at wolfram.com
• Unit-Norm Vectors under Different p-Norms at wolfram.com
• Simple IEEE 754 implementation in C++
• NPM JavaScript Package/Module