Hemimetrics

Hemimetrics are distance-like functions that generalize metrics by relaxing the symmetry requirement. Formally, a function d: X × X → [0, ∞) on a set X is a hemimetric if it satisfies nonnegativity (d(x, y) ≥ 0 for all x, y), the identity on the diagonal (d(x, x) = 0 for all x), and the triangle inequality (d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in X). Unlike metrics, symmetry is not required; in general, d(x, y) may differ from d(y, x).

If, in addition, symmetry holds for all pairs (i.e., d(x, y) = d(y, x) for all x, y), then the hemimetric is a metric. In some literatures, when the triangle inequality and nonnegativity hold but symmetry is not assumed, the term “quasi-metric” or “asymmetric metric” is used; terminology varies among authors.

Examples help clarify the concept. On the real line, define d(x, y) = max(0, y − x). This function is a hemimetric: it is nonnegative, satisfies d(x, x) = 0, and obeys the triangle inequality, but it is not symmetric since d(y, x) = max(0, x − y). Another natural example is the directed shortest-path distance in a directed graph with nonnegative edge weights: the distance from u to v is nonnegative, zero when u = v, and satisfies the triangle inequality, yet d(u, v) may differ from d(v, u).

Hemimetrics induce non-symmetric topologies and can be symmetrized by standard constructions (for example, taking the maximum or the sum of d(x, y) and d(y, x)). They appear in computer science, optimization, and the study of directed or asymmetric phenomena. See also quasi-metric, metric, and directed-graph distances.