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 (0,0) and (1,1) is , but the point (0,1) is at a distance 1 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 P and Q.
The following figure shows unit circles (the set of all points that are at the unit distance from the centre) with various values of :