Diameter of a set

Definition

Let $(X, d)$ be a metric space and let $S$ be a nonempty subset of $X$. The diameter of $S$ is $$\operatorname{diam}(S) = \sup \{ d(x, y) : x, y \in S \},$$ where the supremum is taken in the extended real number line, so that $\operatorname{diam}(S) \in [0, \infty]$.

Remarks

The set $S$ is bounded if and only if $\operatorname{diam}(S) < \infty$. For the empty set, a common convention is $\operatorname{diam}(\emptyset) = 0$. In $\mathbb{R}$, the interval $(a, b)$ with $a < b$ has diameter $b - a$, and so does $[a, b]$. If $S \subseteq T$, then $\operatorname{diam}(S) \le \operatorname{diam}(T)$.

Explanations

L1 human

The diameter is the largest distance between two points of the set, or the limit of such distances. For a disc of radius $r$ in the plane, the diameter is $2r$.