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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. ==Gromov–Hausdorff distance== 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 ''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. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gromov–Hausdorff convergence」の詳細全文を読む スポンサード リンク
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