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.
Minkowski distance is typically used with p being 1 or 2. The latter is the Euclidean distance, while the former is sometimes known as the Manhattan distance. 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 with various values of p: