For $p\geq 1,$ the Minkowski distance is a metric as a result of the Minkowski inequality. When $p<1,$ the distance between $(0,0)$ and $(1,1)$ is $2^{1/p}>2,$ but the point $(0,1)$ is at a distance $1$ from both of these points. Since this violates the triangle inequality, for $p<1$ it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of $1/p.$ 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:

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 level set of the distance function where all points are at the unit distance from the center) with various values of $p$:

See also

Generalized mean – N-th root of the arithmetic mean of the given numbers raised to the power n

$L^{p))$ space – Function spaces generalizing finite-dimensional p norm spaces