information:extraction, structural:scholar_edition, stylistic:tone
Semimetrics

A semimetric on X {\displaystyle X} is a function d: X × X → R {\displaystyle d:X\times X\to \mathbb {R} } that satisfies the first three axioms, but not necessarily the triangle inequality:

Some authors work with a weaker form of the triangle inequality, such as:

The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and the ρ-relaxed triangle inequality implies the 2ρ-inframetric inequality. Semimetrics satisfying these equivalent conditions have sometimes been referred to as quasimetrics, nearmetrics or inframetrics.

The ρ-inframetric inequalities were introduced to model round-trip delay times in the internet. The triangle inequality implies the 2-inframetric inequality, and the ultrametric inequality is exactly the 1-inframetric inequality.