Minkowski distance

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.

Definition

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

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

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

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

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

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

Similarly, for ${\displaystyle p}$ reaching negative infinity, we have:

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

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

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 ${\displaystyle p}$:

See also

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

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Lp spaces

Basic concepts

• Banach & Hilbert spaces
• L^p spaces
• Measure
  • Lebesgue
• Measure space
• Measurable space/function
• Minkowski distance
• Sequence spaces

L¹ spaces

• Integrable function
• Lebesgue integration
• Taxicab geometry

L² spaces

• Bessel's
• Cauchy–Schwarz
• Euclidean distance
• Hilbert space
• Parseval's identity
• Polarization identity
• Pythagorean theorem
• Square-integrable function

${\displaystyle L^{\infty }}$ spaces

• Bounded function
• Chebyshev distance
• Infimum and supremum
  • Essential
• Uniform norm

Maps

• Almost everywhere
• Convergence almost everywhere
• Convergence in measure
• Function space
• Integral transform
• Locally integrable function
• Measurable function
• Symmetric decreasing rearrangement

Inequalities

• Babenko–Beckner
• Chebyshev's
• Clarkson's
• Hanner's
• Hausdorff–Young
• Hölder's
• Markov's
• Minkowski
• Young's convolution

Results

• Marcinkiewicz interpolation theorem
• Plancherel theorem
• Riemann–Lebesgue
• Riesz–Fischer theorem
• Riesz–Thorin theorem

For Lebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

• Bochner space
• Fourier analysis
• Lorentz space
• Probability theory
• Quasinorm
• Real analysis
• Sobolev space
• *-algebra
  • C*-algebra
  • Von Neumann