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Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces

Nicolò Zava

Algebraic & Geometric Topology 25 (2025) 5153–5174
Bibliography
1 H Adams, J Bush, N Clause, F Frick, M Gómez, M Harrison, R A Jeffs, E Lagoda, S Lim, F Mémoli, M Moy, N Sadovek, M Superdock, D Vargas, Q Wang, L Zhou, Gromov–Hausdorff distances, Borsuk–Ulam theorems, and Vietoris–Rips complexes, preprint (2023) arXiv:2301.00246
2 P K Agarwal, K Fox, A Nath, A Sidiropoulos, Y Wang, Computing the Gromov–Hausdorff distance for metric trees, ACM Trans. Algorithms 14 (2018) 24 MR3814320
3 S A Antonyan, The Gromov–Hausdorff hyperspace of a Euclidean space, Adv. Math. 363 (2020) 106977 MR4052556
4 S A Antonyan, The Gromov–Hausdorff hyperspace of a Euclidean space, II, Adv. Math. 393 (2021) 108055 MR4333735
5 P Assouad, Plongements lipschitziens dans n, Bull. Soc. Math. France 111 (1983) 429 MR763553
6 D Bate, A L Garcia Pulido, Bi-Lipschitz embeddings of the space of unordered m-tuples with a partial transportation metric, Math. Ann. 390 (2024) 3109 MR4801849
7 G Bell, A Dranishnikov, Asymptotic dimension, Topology Appl. 155 (2008) 1265 MR2423966
8 G S Bloom, A counterexample to a theorem of S Piccard, J. Combinatorial Theory Ser. A 22 (1977) 378 MR439657
9 L M Blumenthal, Theory and applications of distance geometry, Clarendon (1953) MR54981
10 M G Bouligand, Ensembles impropres et nombre dimensionnel, Bull. Sci. Math. 52 (1928) 320, 361
11 P Bubenik, A Wagner, Embeddings of persistence diagrams into Hilbert spaces, J. Appl. Comput. Topol. 4 (2020) 339 MR4130975
12 D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
13 G Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009) 255 MR2476414
14 G Carlsson, F Mémoli, Persistent clustering and a theorem of J Kleinberg, preprint (2008) arXiv:0808.2241
15 G Carlsson, F Mémoli, Characterization, stability and convergence of hierarchical clustering methods, J. Mach. Learn. Res. 11 (2010) 1425 MR2645457
16 M Carrière, U Bauer, On the metric distortion of embedding persistence diagrams into separable Hilbert spaces, from: "35th International Symposium on Computational Geometry" (editors G Barequet, Y Wang), Leibniz Int. Proc. Inform. 129, Schloss Dagstuhl (2019) 21 MR3968607
17 F Chazal, D Cohen-Steiner, L J Guibas, F Mémoli, S Y Oudot, Gromov–Hausdorff stable signatures for shapes using persistence, Computer Graphics Forum 28 (2009) 1393
18 F Chazal, V de Silva, S Oudot, Persistence stability for geometric complexes, Geom. Dedicata 173 (2014) 193 MR3275299
19 S Chowdhury, F Mémoli, A functorial Dowker theorem and persistent homology of asymmetric networks, J. Appl. Comput. Topol. 2 (2018) 115 MR3873182
20 S Chowdhury, F Mémoli, Distances and isomorphism between networks : stability and convergence of network invariants, J. Appl. Comput. Topol. 7 (2023) 243 MR4588140
21 G David, M Snipes, A constructive proof of the Assouad embedding theorem with bounds on the dimension, preprint (2012) arXiv:1211.3223
22 A N Dranishnikov, G Gong, V Lafforgue, G Yu, Uniform embeddings into Hilbert space and a question of Gromov, Canad. Math. Bull. 45 (2002) 60 MR1884134
23 J Dydak, Ž Virk, Preserving coarse properties, Rev. Mat. Complut. 29 (2016) 191 MR3438031
24 J Dydak, T Weighill, Monotone-light factorizations in coarse geometry, Topology Appl. 239 (2018) 160 MR3777332
25 H Edelsbrunner, J L Harer, Computational topology: an introduction, Amer. Math. Soc. (2010) MR2572029
26 H Edelsbrunner, T Heiss, V Kurlin, P Smith, M Wintraecken, The density fingerprint of a periodic point set, from: "37th International Symposium on Computational Geometry" (editors K Buchin, É Colin de Verdière), Leibniz Int. Proc. Inform. 189, Schloss Dagstuhl (2021) 32 MR4287039
27 D A Edwards, The structure of superspace, from: "Studies in topology" (editors N M Stavrakas, K R Allen), Academic (1975) 121 MR401069
28 J M Fraser, Assouad dimension and fractal geometry, 222, Cambridge Univ. Press (2021) MR4411274
29 A Garber, Ž Virk, N Zava, On the metric spaces of lattices and periodic point sets, preprint (2023) arXiv:2310.07594
30 B Giunti, J Lazovskis, B Rieck, DONUT : database of original and non-theoretical uses of topology (2022)
31 M Gromov, Structures métriques pour les variétés riemanniennes, 1, CEDIC (1981) MR682063
32 M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, II" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1 MR1253544
33 K Hess, Topological adventures in neuroscience, from: "Topological data analysis — the Abel Symposium 2018" (editors N A Baas, G E Carlsson, G Quick, M Szymik, M Thaule), Abel Symp. 15, Springer (2020) 277 MR4338677
34 N Higson, J Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000) 143 MR1739727
35 S Iliadis, A O Ivanov, A A Tuzhilin, Local structure of Gromov–Hausdorff space, and isometric embeddings of finite metric spaces into this space, Topology Appl. 221 (2017) 393 MR3624471
36 N J Kalton, M I Ostrovskii, Distances between Banach spaces, Forum Math. 11 (1999) 17 MR1673915
37 J Kucab, M Zarichnyi, On asymptotic power dimension, Topology Appl. 201 (2016) 124 MR3461159
38 T J Laakso, Plane with A-weighted metric not bi-Lipschitz embeddable to N, Bull. London Math. Soc. 34 (2002) 667 MR1924353
39 V Lafforgue, Un renforcement de la propriété (T), Duke Math. J. 143 (2008) 559 MR2423763
40 U Lang, C Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001) 285 MR1866853
41 S Lim, F Mémoli, Z Smith, The Gromov–Hausdorff distance between spheres, Geom. Topol. 27 (2023) 3733 MR4674839
42 S Majhi, J Vitter, C Wenk, Approximating Gromov–Hausdorff distance in Euclidean space, Comput. Geom. 116 (2024) 102034 MR4611642
43 F Mémoli, On the use of Gromov–Hausdorff distances for shape comparison, from: "Eurographics Symposium on Point-Based Graphics" (editors M Botsch, R Pajarola, B Chen, M Zwicker), The Eurographics Association (2007) 81
44 F Mémoli, Gromov–Hausdorff distances in Euclidean spaces, from: "2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops" (2008) 1
45 F Mémoli, Some properties of Gromov–Hausdorff distances, Discrete Comput. Geom. 48 (2012) 416 MR2946454
46 F Mémoli, G Sapiro, Comparing point clouds, from: "SGP ’04 : Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing", ACM (2004) 32
47 F Mémoli, G Sapiro, A theoretical and computational framework for isometry invariant recognition of point cloud data, Found. Comput. Math. 5 (2005) 313 MR2168679
48 A Mitra, Ž Virk, The space of persistence diagrams on n points coarsely embeds into Hilbert space, Proc. Amer. Math. Soc. 149 (2021) 2693 MR4246818
49 A Mitra, Ž Virk, Geometric embeddings of spaces of persistence diagrams with explicit distortions, preprint (2024) arXiv:2401.05298
50 T Miyata, Ž Virk, Dimension-raising maps in a large scale, Fund. Math. 223 (2013) 83 MR3125134
51 A Naor, O Neiman, Assouad’s theorem with dimension independent of the snowflaking, Rev. Mat. Iberoam. 28 (2012) 1123 MR2990137
52 P W Nowak, G Yu, Large scale geometry, European Math. Society (2012) MR2986138
53 P Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989) 1 MR979599
54 P Petersen, Riemannian geometry, 171, Springer (1998) MR1480173
55 N Pritchard, T Weighill, Coarse embeddability of Wasserstein space and the space of persistence diagrams, Discrete Comput. Geom. 74 (2025) 358 MR4961363
56 T M Radul, O Shukel, Functors of finite degree and asymptotic dimension, Mat. Stud. 31 (2009) 204 MR2643476
57 J C Robinson, Dimensions, embeddings, and attractors, 186, Cambridge Univ. Press (2011) MR2767108
58 J Roe, Lectures on coarse geometry, 31, Amer. Math. Soc. (2003) MR2007488
59 F Schmiedl, Computational aspects of the Gromov–Hausdorff distance and its application in non-rigid shape matching, Discrete Comput. Geom. 57 (2017) 854 MR3639607
60 S Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A-weights, Rev. Mat. Iberoamericana 12 (1996) 337 MR1402671
61 A A Tuzhilin, Lectures on Hausdorff and Gromov–Hausdorff distance geometry, preprint (2020) arXiv:2012.00756v1
62 A Wagner, Nonembeddability of persistence diagrams with p > 2 Wasserstein metric, Proc. Amer. Math. Soc. 149 (2021) 2673 MR4246816
63 T Weighill, T Yamauchi, N Zava, Coarse infinite-dimensionality of hyperspaces of finite subsets, Eur. J. Math. 8 (2022) 335 MR4384742
64 G Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. 147 (1998) 325 MR1626745
65 G Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000) 201 MR1728880
66 N Zava, Stability of the q-hyperconvex hull of a quasi-metric space, preprint (2022) arXiv:2208.10619