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.


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  :

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