Minkowski Distance

machine-learning statistics distance-metric

Definition

Minkowski Distance

The Minkowski distance is a generalised metric in Euclidean space that parameterises a family of distances. For two vectors and a parameter , the Minkowski distance is defined as the norm of their difference:

Special Cases

Manhattan Distance (): Corresponds to the norm, measuring distance along grid axes.

Euclidean Distance (): Corresponds to the norm, measuring the direct straight-line distance.

Chebyshev Distance (): Defined by the maximum coordinate difference: .

Properties

The Minkowski distance satisfies the triangle inequality (Minkowski inequality) for all . For values , the triangle inequality is violated, and the function does not constitute a formal metric.