Minkowski distance

The Minkowski distance is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

The Minkowski distance of order p between two points

is defined as:

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 (0,0) and (1,1) is ${\displaystyle 2^{1/p}>2}$, but the point (0,1) is at a distance 1 from both of these points. Since this violates the triangle inequality, for ${\displaystyle p<1}$ it is not a metric.

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