The Minkowski distance of order p 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 p being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of p reaching infinity, we obtain the Chebyshev distance:
Similarly, for p 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 p: