Hausdorff distance measures how two compact subsets of a metric space are far from being equal.
Let X and Y be two compact subsets of a metric space M. Then Hausdorff distance dH(X,Y) is the minimal number r such that closed r-neighborhood of X contains Y and closed r-neighborhood of Y contains X. In other words, if |xy| denotes the distance in M then
Hausdorff distance can be defined the same way for closed non-compact subsets of M, but in this case the distance may take infinite value and the topology of F(M) starts to depend on particular metric on M (not only on its topology). The Hausdorff distance between not closed subsets can be defined as the Hausdorff distance between its closers. It gives a pre-metric on the set of all subsets of M (Hausdorff distance between any two sets and with the same closers is zero).
In Euclidean geometry it is often used an analog of Hausdorff distance up to isometry. Namely let X and Y be two compact figures of a in a Euclidean space, then DH(X,Y) is the minimum of dH(I(X),Y) along all isometries I of Euclidean space . This distance measures how far X and Y are from being isometric.
Gromov-Hausdorff distance measures how two compact metric spaces are far from being isometric. Let X and Y be two compact metric spaces, and then dGH(X,Y) is the minimum of all numbers dH(f(X),g(Y)) for all metric spaces M and all isometric embeddings f:X'M and g:Y'M.
(The isometric embedding here understood in the extrinsic, sense i.e it must preserve all distances, not only infinitesimally small, for example no compact Riemannian manifold admit such embedding into Euclidean space)
Gromov-Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, and what is more important into a topological space.
Pointed Gromov-Hausdorff convergence is an appropriate analog of Gromov-Hausdorff convergence for non-compact spaces.
Let (Xi,pi) be a sequence of locally compact complete length-metric spaces with marked points, it is converging to (Y,p) if for any R>0 the closed R-balls around pi in Xi is converging to the R-ball around p in Y in usual Gromov-Hausdorff sense.
The notion of Gromov-Hausdorff convergence was first used to prove that any discrete group with polynomial growth is almost nilpotent (i.e. it contains a nilpotent subgroup of finite index). The key ingredient in the proof was almost trivial observation that for the Kelly graph of a group with polynomial growth a sequence of rescaling converges in the pointed Gromov-Hausdorff sense. Yet one more simple and very important result is
Gromov's compactness theorem. The set of Riemannian manifolds with Ricci curvature >c and diameter
