Gromov-Hausdorff distance

The Gromov-Hausdorff distance is a metric on the space of all compact metric spaces modulo isometry, providing a rigorous notion of how far two abstract metric spaces are from being identical in their distance structure. Introduced by Mikhail Gromov in 1981, it generalizes the classical Hausdorff distance by allowing isometric embeddings into arbitrary ambient spaces rather than requiring a common container, thereby enabling the comparison of spaces with no prior relationship. The distance is central to metric geometry and geometric analysis, where it underpins compactness theorems, convergence of Riemannian manifolds, and the study of singular limit spaces such as Alexandrov spaces. Its power lies in treating metric spaces as points in their own right — a shift from studying geometry inside spaces to studying the geometry of spaces themselves.