Minkowski distance

Short description: Mathematical metric in normed vector 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 [math]\displaystyle{ p }[/math] (where [math]\displaystyle{ p }[/math] is an integer) between two points [math]\displaystyle{ X = (x_1,x_2,\ldots,x_n) \text{ and } Y = (y_1,y_2,\ldots,y_n) \in \R^n }[/math] is defined as: [math]\displaystyle{ D\left(X,Y\right) = \biggl(\sum_{i=1}^n |x_i-y_i|^p\biggr)^{\frac{1}{p}}. }[/math]

For [math]\displaystyle{ p \geq 1, }[/math] the Minkowski distance is a metric as a result of the Minkowski inequality. When [math]\displaystyle{ p \lt 1, }[/math] the distance between [math]\displaystyle{ (0, 0) }[/math] and [math]\displaystyle{ (1, 1) }[/math] is [math]\displaystyle{ 2^{1/p} \gt 2, }[/math] but the point [math]\displaystyle{ (0, 1) }[/math] is at a distance [math]\displaystyle{ 1 }[/math] from both of these points. Since this violates the triangle inequality, for [math]\displaystyle{ p \lt 1 }[/math] it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of [math]\displaystyle{ 1/p. }[/math] The resulting metric is also an F-norm.

Minkowski distance is typically used with [math]\displaystyle{ p }[/math] being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of [math]\displaystyle{ p }[/math] reaching infinity, we obtain the Chebyshev distance: [math]\displaystyle{ \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|. }[/math]

Similarly, for [math]\displaystyle{ p }[/math] reaching negative infinity, we have: [math]\displaystyle{ \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|. }[/math]

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q. }[/math]

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 [math]\displaystyle{ p }[/math]:

See also

• Generalized mean – N-th root of the arithmetic mean of the given numbers raised to the power n
• [math]\displaystyle{ L^p }[/math] space – Function spaces generalizing finite-dimensional p norm spaces
• Norm (mathematics) – Length in a vector space

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