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One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces.
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Gromov M. Metric structures for Riemannian and non-Riemannian spaces , with Appendices by M. Katz, P. Pansu, and S. Semmes. Boston, MA: Birkhauser, 2007
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Latremoliere F. The Quantum Gromov-Hausdorff Propinquity . ArXiv e-prints, arXiv:1302.4058, 2013
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