afstandsfunktion

An afstandsfunktion, commonly called a distance function or metric, is a function that assigns a nonnegative real value to any pair of elements from a set, measuring how far apart they are. Formally, it is a function d: X × X → [0, ∞) that is intended to quantify the distance between elements x and y in X.

A distance function typically satisfies four core properties: non-negativity (d(x,y) ≥ 0), identity of indiscernibles (d(x,y) = 0 if and only if x = y), symmetry (d(x,y) = d(y,x)), and the triangle inequality (d(x,z) ≤ d(x,y) + d(y,z)) for all x, y, z in X. When all these hold, the distance induces a metric space, which provides a robust framework for geometry, analysis, and topology. If the identity of indiscernibles is relaxed so that d(x,y) = 0 does not necessarily imply x = y, the function is called a pseudometric. An ultrametric is a stronger variant where d(x,z) ≤ max{d(x,y), d(y,z)}.

Common examples include the Euclidean distance (L2 norm), the Manhattan distance (L1 norm), and the Chebyshev distance (L∞ norm), as well as Minkowski distances for general p ≥ 1. In graphs, distances can be defined by the length of the shortest path with weighted edges.

A distance function induces a topology via open balls and supports notions of convergence and continuity. Not all distance-like measures are true metrics; some divergences, such as the Kullback–Leibler divergence, are not symmetric or do not satisfy the triangle inequality but are still used to compare objects in information theory and statistics.