Minkowski distance is a distance/ similarity measurement between two points in the normed vector space (N dimensional real space) and is a generalization of the Euclidean distance and the Manhattan distance.

Minkowski distance in N-D space

In a N dimensional space, a point is represented as (x1, x2, ..., xN).

Consider two points P1 and P2:

P1: (X1, X2, ..., XN)
P2: (Y1, Y2, ..., YN)

Then, the Minkowski distance between P1 and P2 is given as:

$$ \sqrt[p]{{(x1-y1)}^p\ +\ {(x2-y2)}^p\ +\ ...\ +\ {(xN-yN)}^p}
$$

Euclidean distance from Minkowski distance

When p = 2, Minkowshi distance is same as Euclidean distance.

Manhattan distance from Minkowski distance

When p = 1, Minkowshi distance is same as Manhattan distance.

Visualize

Unit circles (path represents points with same Minkowshi distance) with various values of p (Minkowski distance):

Applications

Applications of Minkowshi Distance are:

• Fuzzy Clustering with Minkowski Distance Functions

• A Framework for a Minkowski Distance Based Multi Metric Quality of Service Monitoring Infrastructure for Mobile Ad Hoc Networks

Euclidean distance and Manhattan distance are same as Minkowshi Distance, hence, the applications of the previous two distance metrics are applications of Minkowshi Distance.