Diameter

The diameter of a subset $A$ of a metric space $(X,d)$ is

where $\sup$ denotes the supremum (and the value may be $+\infty$). By convention, $\operatorname{diam}(\varnothing)=0$.

Diameter measures the “size” of a set in a way that is tailored to the metric ; it is directly tied to the notion of a bounded set .

Examples:

• In $(\mathbb{R},|\cdot|)$, $\operatorname{diam}([a,b])=b-a$ for $a\le b$.
• In $\mathbb{R}^2$ with the Euclidean metric, the unit circle has diameter $2$.