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The Gromov–Hausdorff distance metric has been applied in the field of computer graphics and computational geometry to find correspondences between different shapes. It also has been applied in the problem of
183:
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
187:
of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Gromov–Hausdorff limit of the sequence.
215:. In the global sense, the Gromov–Hausdorff space is totally heterogeneous, i.e., its isometry group is trivial, but locally there are many nontrivial isometries.
223:
The pointed Gromov–Hausdorff convergence is an analog of Gromov–Hausdorff convergence appropriate for non-compact spaces. A pointed metric space is a pair (
757:
Sukkar, Fouad; Wakulicz, Jennifer; Lee, Ki Myung Brian; Fitch, Robert (2022-09-11). "Motion planning in task space with Gromov-Hausdorff approximations".
321:
541:
Ivanov, A. O.; Nikolaeva, N. K.; Tuzhilin, A. A. (2016). "The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic".
382:
to prove the stability of the
Friedmann model in Cosmology. This model of cosmology is not stable with respect to smooth variations of the metric.
423:
629:
Ivanov, Alexander O.; Tuzhilin, Alexey A. (2016). "Local
Structure of Gromov-Hausdorff Space near Finite Metric Spaces in General Position".
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in the Gromov–Hausdorff metric. The limit spaces are metric spaces. Additional properties on the length spaces have been proven by
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is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact
667:
650:
Bellaïche, André (1996). "The tangent space in sub-Riemannian geometry". In André Bellaïche; Jean-Jacques Risler (eds.).
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The Gromov–Hausdorff distance was introduced by David
Edwards in 1975, and it was later rediscovered and generalized by
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864:
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Kotani, Motoko; Sunada, Toshikazu (2006). "Large deviation and the tangent cone at infinity of a crystal lattice".
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60:
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324:. (Also see D. Edwards for an earlier work.) The key ingredient in the proof was the observation that for the
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of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense.
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David A. Edwards, "The
Structure of Superspace", in "Studies in Topology", Academic Press, 1975,
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386:
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Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on
Geometry processing - SGP '04
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M. Gromov. "Structures métriques pour les variétés riemanniennes", edited by
Lafontaine and
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204:
587:
Chowdhury, Samir; Mémoli, Facundo (2016). "Explicit
Geodesics in Gromov-Hausdorff Space".
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8:
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Ivanov, Alexander; Tuzhilin, Alexey (2018). "Isometry Group of Gromov--Hausdorff Space".
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475:"Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits)"
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In a special case, the concept of Gromov–Hausdorff limits is closely related to
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The notion of Gromov–Hausdorff convergence was used by Gromov to prove that any
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654:. Progress in Mathematics. Vol. 44. Basel: Birkhauser. pp. 1–78 .
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Tuzhilin, Alexey A. (2016). "Who
Invented the Gromov-Hausdorff Distance?".
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How far and how near are some figures under the Gromov–Hausdorff distance.
20:
490:
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Sormani, Christina (2004). "Friedmann cosmology and almost isotropy".
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Mémoli, Facundo; Sapiro, Guillermo (2004). "Comparing point clouds".
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685:"On the structure of spaces with Ricci curvature bounded below. I"
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254:) of pointed metric spaces converges to a pointed metric space (
211:, i.e., any two of its points are the endpoints of a minimizing
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Metric structures for
Riemannian and non-Riemannian spaces
339:, which states that the set of Riemannian manifolds with
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218:
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585:For explicit construction of the geodesics, see
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378:The Gromov–Hausdorff distance has been used by
322:Gromov's theorem on groups of polynomial growth
682:
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822:
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63:in 1981. This distance measures how far two
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312:is virtually nilpotent (i.e. it contains a
683:Cheeger, Jeff; Colding, Tobias H. (1997).
266: > 0, the sequence of closed
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331:Another simple and very useful result in
191:Some properties of Gromov–Hausdorff space
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873:
867:(translation with additional content).
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296:in the usual Gromov–Hausdorff sense.
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479:Publications Mathématiques de l'IHÉS
219:Pointed Gromov–Hausdorff convergence
79:are two compact metric spaces, then
123:)) for all (compact) metric spaces
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527:D. Burago, Yu. Burago, S. Ivanov,
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780:Geometric and Functional Analysis
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195:The Gromov–Hausdorff space is
176:admits such an embedding into
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127:and all isometric embeddings
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39:which is a generalization of
337:Gromov's compactness theorem
25:Gromov–Hausdorff convergence
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660:10.1007/978-3-0348-9210-0_1
529:A Course in Metric Geometry
395:
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891:Convergence (mathematics)
837:10.1007/s00209-006-0951-9
825:Mathematische Zeitschrift
802:10.1007/s00039-004-0477-4
565:10.1134/S0001434616110298
47:Gromov–Hausdorff distance
473:Gromov, Michael (1981).
284:converges to the closed
726:10.1145/1057432.1057436
652:Sub-Riemannian Geometry
402:Intrinsic flat distance
387:large-deviations theory
180:of the same dimension.
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859:, Birkhäuser (1999).
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886:Riemannian geometry
531:, AMS GSM 33, 2001.
333:Riemannian geometry
174:Riemannian manifold
170:isometric embedding
164:between subsets in
543:Mathematical Notes
491:10.1007/BF02698687
426:2016-03-04 at the
357:relatively compact
314:nilpotent subgroup
162:Hausdorff distance
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29:Mikhail Gromov
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316:of finite
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