63:
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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
194:
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
332:
198:
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.
226:. 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.
234:
The pointed Gromov–Hausdorff convergence is an analog of Gromov–Hausdorff convergence appropriate for non-compact spaces. A pointed metric space is a pair (
768:
Sukkar, Fouad; Wakulicz, Jennifer; Lee, Ki Myung Brian; Fitch, Robert (2022-09-11). "Motion planning in task space with Gromov-Hausdorff approximations".
219:
347:
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Ivanov, A. O.; Nikolaeva, N. K.; Tuzhilin, A. A. (2016). "The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic".
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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.
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678:
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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
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39:
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Bellaïche, André (1996). "The tangent space in sub-Riemannian geometry". In André Bellaïche; Jean-Jacques Risler (eds.).
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70:
The Gromov–Hausdorff distance was introduced by David
Edwards in 1975, and it was later rediscovered and generalized by
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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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891:
335:. (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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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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Chowdhury, Samir; Mémoli, Facundo (2016). "Explicit
Geodesics in Gromov-Hausdorff Space".
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Ivanov, Alexander; Tuzhilin, Alexey (2018). "Isometry Group of Gromov--Hausdorff Space".
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486:"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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712:
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736:
665:. 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.
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79:
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106:
789:
Sormani, Christina (2004). "Friedmann cosmology and almost isotropy".
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Mémoli, Facundo; Sapiro, Guillermo (2004). "Comparing point clouds".
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696:"On the structure of spaces with Ricci curvature bounded below. I"
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265:) of pointed metric spaces converges to a pointed metric space (
222:, 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
350:, which states that the set of Riemannian manifolds with
596:For explicit construction of the geodesics, see
389:The Gromov–Hausdorff distance has been used by
333:Gromov's theorem on groups of polynomial growth
8:
74:in 1981. This distance measures how far two
323:is virtually nilpotent (i.e. it contains a
694:Cheeger, Jeff; Colding, Tobias H. (1997).
277: > 0, the sequence of closed
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773:
711:
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624:
603:
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342:Another simple and very useful result in
202:Some properties of Gromov–Hausdorff space
423:
27:Notion for convergence of metric spaces
878:(translation with additional content).
307:in the usual Gromov–Hausdorff sense.
7:
490:Publications Mathématiques de l'IHÉS
230:Pointed Gromov–Hausdorff convergence
90:are two compact metric spaces, then
134:)) for all (compact) metric spaces
538:D. Burago, Yu. Burago, S. Ivanov,
25:
791:Geometric and Functional Analysis
46:, is a notion for convergence of
700:Journal of Differential Geometry
242:) consisting of a metric space
206:The Gromov–Hausdorff space is
187:admits such an embedding into
1:
138:and all isometric embeddings
78:metric spaces are from being
50:which is a generalization of
348:Gromov's compactness theorem
36:Gromov–Hausdorff convergence
18:Gromov-Hausdorff convergence
671:10.1007/978-3-0348-9210-0_1
540:A Course in Metric Geometry
918:
902:Convergence (mathematics)
848:10.1007/s00209-006-0951-9
836:Mathematische Zeitschrift
813:10.1007/s00039-004-0477-4
576:10.1134/S0001434616110298
58:Gromov–Hausdorff distance
484:Gromov, Michael (1981).
295:converges to the closed
737:10.1145/1057432.1057436
663:Sub-Riemannian Geometry
413:Intrinsic flat distance
398:large-deviations theory
191:of the same dimension.
105:) is defined to be the
713:10.4310/jdg/1214459974
67:
870:, Birkhäuser (1999).
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897:Riemannian geometry
542:, AMS GSM 33, 2001.
344:Riemannian geometry
185:Riemannian manifold
181:isometric embedding
175:between subsets in
554:Mathematical Notes
502:10.1007/BF02698687
437:2016-03-04 at the
368:relatively compact
325:nilpotent subgroup
173:Hausdorff distance
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52:Hausdorff distance
680:978-3-0348-9946-8
321:polynomial growth
16:(Redirected from
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44:Felix Hausdorff
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72:Mikhail Gromov
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40:Mikhail Gromov
38:, named after
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386:in robotics.
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866:M. Gromov.
32:mathematics
886:Categories
775:2209.04800
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626:1806.02100
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419:References
327:of finite
246:and point
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216:separable
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80:isometric
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360:diameter
224:geodesic
220:geodesic
212:complete
179:and the
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