---
tag: 0692CGQF00
kind: definition
name: Diameter of a set
msc: ["54E35"]
defines: 0KW3PN547K
uses: [063W5JMY85, 0BTMQ04556, 0GSZ7B0CGR]
sources:
  - url: https://en.wikipedia.org/wiki/Diameter
    locator: Generalizations
---

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] 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|0BTMQ04556]] is taken in the [[extended real number line|0GSZ7B0CGR]], so that $\operatorname{diam}(S) \in [0, \infty]$.

## Remarks

The set $S$ is [[bounded|0P953MG05E]] 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 $2r$.