A metric-space statement that for any three points x, y, z the distance from x to z does not exceed the sum of the distances from x to y and from y to z; symbolically d(x,z) ≤ d(x,y)+d(y,z).

Principle

The organizing rule that distances in a metric obey subadditivity: taking a detour via any intermediate point cannot shorten the total distance below the direct distance.

Demonstration

In the Euclidean plane take points A=(0,0), B=(1,0), C=(1,1). Distances: AB=1, BC=1, AC=√2; indeed AC ≤ AB+BC since √2 ≤ 2. In normed vector spaces the inequality is ||u+v|| ≤ ||u||+||v|| for vectors, which is equivalent to the triangle inequality for the induced metric.

Misapplication

Applying the triangle inequality to signed or directed distances without accounting for orientation, or assuming it holds for arbitrary similarity scores or dissimilarity measures that are not metrics.

Consequence

It guarantees that straight-line (geodesic) segments are shortest paths in metric spaces, underpins convergence arguments in analysis, and ensures norm-induced topologies behave predictably.

Reversal

Negating the inequality (d(x,z) ≥ d(x,y)+d(y,z)) is false in general; a meaningful contrast is an ultrametric (non-Archimedean) condition where the stronger inequality d(x,z) ≤ max{d(x,y),d(y,z)} holds instead of the usual additive bound.

Boundary

Applies to metrics and pseudometrics (nonnegative, symmetric, satisfying triangle inequality). It does not apply to arbitrary binary relations called ‘distances’ that lack one of the metric axioms, nor to measures that can take negative values.

Semantic Tension

Tension exists between the standard triangle inequality (additive bound) and the ultrametric property (maximum bound); some results or intuitions valid under the usual inequality fail or simplify under the ultrametric case.

Synthesis

The triangle inequality is the subadditivity axiom of metric and normed spaces: it prevents shortcuts that violate distance, supports the geometry of shortest paths, and is a minimal structural rule from which many analytic and topological consequences follow.