Gromov–Hausdorff convergence

In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then d_GH (X, Y) is defined to be the infimum of all numbers d_H(f(X), g(Y)) for all metric spaces M and all isometric embeddings f: X→M and g: Y→M. Here d_H denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension.

The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, called Gromov–Hausdorff space, and it therefore defines a notion of convergence for sequences of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Hausdorff limit of the sequence.

The Gromov–Hausdorff space is path-connected, complete, and separable. It was also shown that it is geodesic, i.e., any two of its points are the endpoints of a minimizing geodesic. Explicit geodesics were constructed in

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