---
tag: TS5XH58X86
kind: definition
name: Gromov–Hausdorff convergence
msc: [53C23, "54E35"]
defines: GT56RTG9X5
uses: [ETRF2FHAEE, 0VKD1TP7F3, 0TVJZYB6SH]
sources:
  - url: https://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence
---

## Definition

Let $X_1, X_2, \ldots$ and $X$ be nonempty [[compact|0TVJZYB6SH]] [[metric spaces|0VKD1TP7F3]].
Let $d_{GH}$ be the Gromov–Hausdorff distance, the metric of the [[Gromov–Hausdorff space|ETRF2FHAEE]].
The sequence $(X_n)$ *converges to $X$ in the Gromov–Hausdorff sense* if $d_{GH}(X_n, X) \to 0$ as $n \to \infty$.

## Remarks

Example: let $X_n = \set{k/n : k = 0, 1, \ldots, n}$ and $X = [0, 1]$, both with the distance $\abs{s - t}$.
Both lie in $\R$, and each point of $[0, 1]$ is within $\tfrac{1}{2n}$ of $X_n$.
So $d_{GH}(X_n, X) \le \tfrac{1}{2n} \to 0$, and $X_n$ converges to $[0, 1]$.

Non-example: let $X_n$ be the round sphere of radius $n$ with its great-circle distance.
One has $\abs{\operatorname{diam} Y - \operatorname{diam} Z} \le 2\, d_{GH}(Y, Z)$ for compact $Y$ and $Z$.
The diameters $\pi n$ tend to infinity, so $(X_n)$ converges to no compact space.

The limit is unique up to isometry, because $d_{GH}$ is a metric on isometry classes.
For spaces that are not compact, such as complete Riemannian manifolds, one uses *pointed* Gromov–Hausdorff convergence.
There a base point is fixed in each space, and one asks for convergence of the closed balls of each radius $r$ around the base points.