Minkowski distance
The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after the German mathematician Hermann Minkowski. Definition. The Minkowski distance of order formula_1 (where formula_1 is an integer) between two points formula_3 is defined as: formula_4 For formula_5 the Minkowski distance is a metric as a result of the Minkowski inequality. When formula_6 the distance between formula_7 and formula_8 is formula_9 but the point formula_10 is at a distance formula_11 from both of these points . Since this violates the triangle inequality, for formula_12 it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of formula_13 The resulting metric is also an F-norm. Minkowski distance is typically used with formula_1 being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of formula_1 reaching infinity, we obtain the Chebyshev distance: formula_16 Similarly, for formula_1 reaching negative infinity, we have: formula_18 The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise  differences between formula_19 and formula_20 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 formula_1: