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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


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

is defined as:

For the Minkowski distance is a metric as a result of the Minkowski inequality. When the distance between and is but the point is at a distance 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 being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of reaching infinity, we obtain the Chebyshev distance:

Similarly, for 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 and

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 :

Unit circles using different Minkowski distance metrics.

See also