Triangle inequality

The triangle inequality states that for any three points $x, y, z$ in a metric space $(X, d)$:

This is one of the defining axioms of a metric.

Equivalent forms

For vectors in a normed space :

For real or complex numbers:

Consequences

• Reverse triangle inequality: $\bigl| d(x,y) - d(y,z) \bigr| \leq d(x,z)$.
• Polygon inequality: $d(x_1, x_n) \leq \sum_{i=1}^{n-1} d(x_i, x_{i+1})$.

The name comes from Euclidean geometry: the length of one side of a triangle is at most the sum of the other two sides.