Definition:Metric Space/Triangle Inequality

Definition

Let $M = \struct {A, d}$ be a metric space, satisfying the metric space axioms:

\((\text M 1)\)	$:$			\(\ds \forall x \in A:\)	\(\ds \map d {x, x} = 0 \)
\((\text M 2)\)	$:$		Triangle Inequality:	\(\ds \forall x, y, z \in A:\)	\(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)
\((\text M 3)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds \map d {x, y} = \map d {y, x} \)
\((\text M 4)\)	$:$			\(\ds \forall x, y \in A:\)	\(\ds x \ne y \implies \map d {x, y} > 0 \)

Axiom $\text M 2$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.

Examples

$4$ Points

Let $M = \struct {A, d}$ be a metric space.

Let $x, y, z, t \in A$.

Then:

$\map d {x, z} + \map d {y, t} \ge \size {\map d {x, y} - \map d {z, t} }$

Also see

• Reverse Triangle Inequality:

$\forall x, y, z \in X: \size {\map d {x, z} - \map d {y, z} } \le \map d {x, y}$

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): triangle inequality: 2.
• 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): One: Metric Spaces: $1$: Open and Closed Sets: $(1.4)$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): triangle inequality