submetric

Submetric is a term encountered in mathematics and related fields with no single universally accepted definition; its meaning depends on the context. In its most straightforward use, a submetric refers to the restriction of a metric to a subset of its domain. If (X, d) is a metric and S is a subset of X, then the restriction of d to S × S is a metric on S, sometimes described as a submetric on S. It preserves nonnegativity, definiteness, symmetry, and the triangle inequality within S.

In other contexts, submetric may denote a distance-like function that is dominated by a metric, i.e., for all x, y one has δ(x, y) ≤ d(x, y), where d is a metric on the same set. Such a function preserves some metric properties but may not satisfy all metric axioms, depending on the exact definition used by authors.

Submetric concepts are closely related to standard generalized distance notions such as pseudometrics (where distinct points can have zero distance), semimetrics, and quasi-metrics (where some axioms are modified). In applied and computational settings, submetrics frequently arise when working with subspaces, restricted domains, imprecise measurements, or approximation schemes.

Because there is no universal formal definition, it is important to review the specific axioms or properties assumed in a given discussion of submetrics.

See also: metric space, pseudometric, quasi-metric, semimetric, restricted metric.