Gromov–Hausdorff c
Gromov–Hausdorff distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then dGH (X,Y ) is defined to be the infimum of all numbers dH(f (X ), g (Y )) for all metric spaces M and all isometric embeddings f :X→M and g :Y→M. Here dH 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 of negative sectional curvature admits such an embedding into Euclidean space.
