L1Distanzen

L1Distanzen, also known as Manhattan distance or taxicab geometry, is a metric in a vector space. It is defined as the sum of the absolute differences of the coordinates of two points. For two points p and q in an n-dimensional space, the L1 distance is given by the formula: d1(p, q) = sum(|pi - qi|) for i from 1 to n. This metric is named after the grid-like street layout of Manhattan, where a taxi would have to travel along the streets rather than directly through city blocks.

Unlike the more common Euclidean distance (L2 distance), which represents the straight-line distance between two points, the L1 distance measures distance by summing the absolute differences along each dimension. This makes it a more appropriate choice in certain applications where movement is restricted to orthogonal directions, such as in city navigation or in some machine learning algorithms.

The L1 distance is always non-negative and satisfies the properties of a metric: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. It is widely used in various fields, including computer science, statistics, and operations research. For example, it is used in feature selection in machine learning, image processing, and in calculating the difference between probability distributions. In essence, the L1 distance provides a way to quantify dissimilarity between two vectors by considering the cumulative deviation across all their components.