Minkowskitávolság

Minkowskitávolság, also known as Minkowski distance or L_p distance, is a metric in a normed vector space. It is defined by the Minkowski inequality, which states that for any two vectors x and y, the distance between them is the p-norm of their difference. This can be expressed mathematically as:

d_p(x, y) = (sum from i=1 to n of |x_i - y_i|^p)^(1/p)

where x and y are n-dimensional vectors and p is a real number greater than or equal to 1.

The Minkowski distance is a generalization of several other common distance metrics. When p = 1, it becomes the Manhattan distance (also known as L_1 distance or taxicab distance), which calculates the distance between two points by summing the absolute differences of their Cartesian coordinates. When p = 2, it is the familiar Euclidean distance (L_2 distance), which represents the shortest straight-line distance between two points. As p approaches infinity, the Minkowski distance converges to the Chebyshev distance (L_infinity distance), which is the maximum absolute difference between the coordinates of the two points.

The parameter p influences the "shape" of the unit ball. For p = 1, the unit ball is a diamond (a square rotated by 45 degrees). For p = 2, it is a circle. For larger values of p, the unit ball increasingly resembles a square. The Minkowski distance finds applications in various fields, including machine learning, pattern recognition, and data mining, where it is used to quantify the dissimilarity between data points.