The Hausdorff metric h on \ mathcal { K } is defined by
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The proof uses the Gromov Hausdorff metric mentioned above.
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A valuation is called continuous if it is continuous with respect to the Hausdorff metric.
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For the manifolds obtained as quotients this amounts to them being convergent in the pointed Gromov-Hausdorff metric.
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There are many ways to define metrics, and for the sets associated to shapes the Hausdorff metric is another.
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In the 1980s, Gromov introduced the Gromov Hausdorff metric, a measure of the difference between two compact metric spaces.
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In " Principles ", he developed the theory of metric spaces and systematically enriched it through a series of new concepts : Hausdorff metric, complete, total boundedness, \ rho-connectivity, reducible sets.
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For a distance measure between two non-empty sets of points you may consider the Hausdorff metric, which is a proper metric, instead of the wishy-washy measures invented by social scientists .-- Lambiam 00 : 58, 1 March 2008 ( UTC)
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Alexandrov spaces with curvature & ge; " k " are important as they form the limits ( in the Gromov-Hausdorff metric ) of sequences of Riemannian manifolds with sectional curvature & ge; " k ", as described by Gromov's compactness theorem.
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