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Nonpositive curvature is not coarsely universal

Inventiones mathematicae volume 217, pages 833–886 (2019)Cite this article

Abstract

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (Geometric group theory, Cambridge University Press, Cambridge, 1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni et al. (Ann Sci Éc Norm Supér (4) 51(3):657–700, 2018). We establish this statement by proving that a metric space which is q-barycentric for some \(q\in [1,\infty )\) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the \(\ell _\infty \) grid \([m]_\infty ^n=(\{1,\ldots ,m\}^n,\Vert \cdot \Vert _\infty )\) into \(\ell _q\) for all \(q\in (2,\infty )\), from which we deduce the following discrete converse to the fact that \(\ell _\infty ^n\) embeds with distortion O(1) into \(\ell _q\) for \(q=O(\log n)\). A rigidity theorem of Ribe (Ark Mat 14(2):237–244, 1976) implies that for every \(n\in {\mathbb {N}}\) there exists \(m\in {\mathbb {N}}\) such that if \([m]_\infty ^n\) embeds into \(\ell _q\) with distortion O(1), then q is necessarily at least a universal constant multiple of \(\log n\). Ribe’s theorem does not give an explicit upper bound on this m, but by the work of Bourgain (Geometrical aspects of functional analysis (1985/86), Springer, Berlin, 1987) it suffices to take \(m=n\), and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

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Notes

  1. Strictly speaking, the above definition is of Alexandrov spaces of global nonpositive curvature, also known as \(\text {CAT}(0)\) spaces or Hadamard spaces. See [27] for the local counterpart of this definition; we will not treat it here, and therefore it will be convenient to drop the term “global” throughout the present text, because the ensuing results are vacuously false under the weaker local assumption (since the 1-dimensional simplicial complex that is associated to any connected combinatorial graph is an Alexandrov space of local nonpositive curvature). For nonnegative curvature, the local and global notions coincide due to the (metric version of the) Alexandrov–Toponogov theorem; see e.g. [152].

  2. In addition to the usual “\(O(\cdot ),o(\cdot )\)” asymptotic notation, it will be convenient to use throughout this article the following (also standard) asymptotic notation. Given two quantities \(Q,Q'>0\), the notations \(Q\lesssim Q'\) and \(Q'\gtrsim Q\) mean that \(Q\leqslant CQ'\) for some universal constant \(C>0\). The notation \(Q\asymp Q'\) stands for \((Q\lesssim Q') \wedge (Q'\lesssim Q)\). If we need to allow for dependence on parameters, we indicate this by subscripts. For example, in the presence of auxiliary objects (e.g. numbers or spaces) \(\phi ,\mathfrak {Z}\), the notation \(Q\lesssim _{\phi ,\mathfrak {Z}} Q'\) means that \(Q\leqslant C(\phi ,\mathfrak {Z})Q' \), where \(C(\phi ,\mathfrak {Z}) >0\) is allowed to depend only on \(\phi ,\mathfrak {Z}\); similarly for the notations \(Q\gtrsim _{\phi ,\mathfrak {Z}} Q'\) and \(Q\asymp _{\phi ,\mathfrak {Z}} Q'\).

  3. A shorter proof of this fact, using an ultrapower and differentiation argument, follows form [76]. Because for this particular case of Ribe’s theorem the target space is \(\ell _q\), a further simplification of the differentiation step is possible; see also [19, 111].

  4. Specifically, in [71, §15(b)] Gromov wrote “The geodesic property is one logical level up from concentration inequalities as it involves the existential quantifier. It is unclear if there is a simple \(\exists \)-free description of (nongeodesic!) subspaces in \(\text {CAT}(\kappa )\)-spaces.” The term “concentration inequalities” is defined in [71, §15(a)] to be the same inequalities as the quadratic metricinequalities that we consider in (12), except that in [71] they are allowed to involve arbitrary powers of the pairwise distances. However, due to [4] it suffices to consider only quadratic inequalities for the purpose of the simple intrinsic description that Gromov hopes to obtain (though, as he indicates, it may not exist).

  5. This demonstrates that the role of the scaling parameter in the definition of metric cotype is not as subsidiary as it may seem from [121], where it did not have a crucial role in the metric characterization of Rademacher cotype or the nonlinear Maurey–Pisier theorem. While sharp metric cotype was shown in [121] to have implications to coarse and uniform embeddings, there is a definite possibility that metric cotype q and sharp metric cotype q coincide for Banach spaces (though this is a major open problem); here we see that this is markedly not so in the setting of Alexandrov geometry, leading to formidable qualitative differences between the coarse implications of the sign of curvature.

  6. In terms of bi-Lipschitz distortion of arbitrary expanders, Table 1 does not fully indicate the extent to which nonnegative curvature behaves better than nonpositive curvature. The fact [141] that an Alexandrov space of nonnegative curvature has Markov type 2 implies (by examining the standard random walk on the graph) that if \(\mathcal {G}=\{\mathsf {G}(n)\}_{n=1}^\infty \) is an expander, then \(\mathsf {c}_X(\mathsf {G}(n))\gtrsim _{\mathcal {G}} \sqrt{\log |\mathsf {G}(n)|}\) for any nonnegatively curved Alexandrov space X. In contrast, there is [96] an Alexandrov space of nonpositive curvature which contains an expander with O(1) distortion.

  7. The only additional observation that is needed for this is that, because the degree of \(\mathsf {G}(n)\) is O(1), a quick and standard counting argument (see e.g. the justification of equation (36) in [134]) shows that \(d_{\mathsf {G}(n)}(u,v)\gtrsim \log |C_j(n)|\rightarrow \infty \) for a constant fraction of \((u,v)\in C_j(n)\times C_j(n)\).

References

  1. Aharoni, I., Maurey, B., Mityagin, B.S.: Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces. Isr. J. Math. 52(3), 251–265 (1985)

    MathSciNet  MATH  Google Scholar 

  2. Alon, N., Lubotzky, A., Wigderson, A.: Semi-direct product in groups and zig-zag product in graphs: connections and applications (extended abstract). In: 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas. NV, 2001), pp. 630–637. IEEE Computer Soc, Los Alamitos, CA (2001)

  3. Alon, N., Milman, V.D.: \(\lambda _1,\) isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38(1), 73–88 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Andoni, A., Naor, A., Neiman, O.: Snowflake universality of Wasserstein spaces. Ann. Sci. Éc. Norm. Supér. (4) 51(3), 657–700 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Andoni, A., Naor, A., Nikolov, A., Razenshteyn, I., Waingarten, E.: Locality sensitive hashing via nonlinear spectral gaps. In: STOC’18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 787–800. ACM, New York, (2018)

  6. Arzhantseva, G., Delzant, T.: Examples of random groups. To appear in J. Topol. (preprint) (2011)

  7. Arzhantseva, G., Tessera, R.: Relative expanders. Geom. Funct. Anal. 25(2), 317–341 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Austin, T., Naor, A.: On the bi-Lipschitz structure of Wasserstein spaces. Preprint (2017)

  9. Austin, T., Naor, A., Tessera, R.: Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces. Groups Geom. Dyn. 7(3), 497–522 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Awasthi, P., Charikar, M., Krishnaswamy, R., Sinop, A.K.: Spectral embedding of k-cliques, graph partitioning and k-means. In: ITCS’16—Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pp. 301–310. ACM, New York (2016)

  11. Ball, K.: Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal. 2(2), 137–172 (1992)

    MathSciNet  MATH  Google Scholar 

  12. Ball, K.: The Ribe programme. Astérisque, (352):Exp. No. 1047, viii, 147–159, 2013. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058

  13. Ball, K., Carlen, E.A., Lieb, E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115(3), 463–482 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Bartal, Y., Gottlieb, L.: Approximate nearest neighbor search for \(\ell _p\)-spaces (\(2<p<\infty \)) via embeddings. In: Proceedings of LATIN 2018, volume 10807 of Lecture Notes in Computer Science, pp. 120–133. Springer (2018)

  15. Baudier, F.: On the metric geometry of stable metric spaces. arXiv:1409.7738 (2014)

  16. Baudier, F., Lancien, G., Motakis, P., Schlumprecht, T.: A new coarsely rigid class of Banach spaces. arXiv:1806.00702 (2018)

  17. Baudier, F., Lancien, G., Schlumprecht, T.: The coarse geometry of Tsirelson’s space and applications. J. Am. Math. Soc. 31(3), 699–717 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Bačác, M.: Old and new challenges in Hadamard spaces. arXiv:1807.01355 (2018)

  19. Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Volume 48 of American Mathematical Society Colloquium Publications, vol. 1. American Mathematical Society, Providence, RI (2000)

    MATH  Google Scholar 

  20. Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata 133, 195–218 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Berg, I.D., Nikolaev, I.G.: Characterization of Aleksandrov spaces of curvature bounded above by means of the metric Cauchy–Schwarz inequality. Mich. Math. J. 67(2), 289–332 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Bourgain, J.: A Hausdorff–Young inequality for \(B\)-convex Banach spaces. Pac. J. Math. 101(2), 255–262 (1982)

    MathSciNet  MATH  Google Scholar 

  23. Bourgain, J.: The metrical interpretation of superreflexivity in Banach spaces. Isr. J. Math. 56(2), 222–230 (1986)

    MathSciNet  MATH  Google Scholar 

  24. Bourgain, J.: On lattice packing of convex symmetric sets in \({\mathbb{R}}^n\). In: Lindenstrauss, J., Milman, V.D. (eds.) Geometrical aspects of functional analysis (1985/86), volume 1267 of Lecture Notes in Mathematics, pp 5–12. Springer, Berlin (1987)

  25. Bourgain, J.: Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometrical Aspects of Functional Analysis (1985/86), volume 1267 of Lecture Notes in Mathematics, pp. 157–167. Springer, Berlin (1987)

  26. Bourgain, J., Milman, V., Wolfson, H.: On type of metric spaces. Trans. Am. Math. Soc. 294(1), 295–317 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999)

    MATH  Google Scholar 

  28. Brown, N., Guentner, E.: Uniform embeddings of bounded geometry spaces into reflexive Banach space. Proc. Am. Math. Soc. 133(7), 2045–2050 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001)

    MATH  Google Scholar 

  30. Burago, Y., Gromov, M., Perelman, G., Aleksandrov, A.D.: Spaces with curvatures bounded below. Uspekhi Mat. Nauk 47(2(284)), 3–51 (1992)

    MathSciNet  Google Scholar 

  31. Butler, G.J.: Simultaneous packing and covering in euclidean space. Proc. Lond. Math. Soc. 3(25), 721–735 (1972)

    MathSciNet  MATH  Google Scholar 

  32. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp 195–199. Princeton Univ. Press, Princeton, N. J (1970)

  33. Cheng, Q.: Sphere equivalence, property H, and Banach expanders. Studia Math. 233(1), 67–83 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., Valette, A.: Groups with the Haagerup Property. Modern Birkhäuser Classics. Birkhäuser/Springer, Basel (2001). Gromov’s a-T-menability, Paperback reprint of the 2001 edition [ MR1852148]

  35. Christiansen, T., Sturm, K.T.: Expectations and martingales in metric spaces. Stochastics 80(1), 1–17 (2008)

    MathSciNet  MATH  Google Scholar 

  36. Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc. 40(3), 396–414 (1936)

    MathSciNet  MATH  Google Scholar 

  37. de la Salle, M.: Towards strong Banach property (T) for \({\rm SL}(3,{{\mathbb{R}}})\). Isr. J. Math. 211(1), 105–145 (2016)

    MathSciNet  MATH  Google Scholar 

  38. de Laat, T., de la Salle, M.: Strong property (T) for higher-rank simple Lie groups. Proc. Lond. Math. Soc. (3) 111(4), 936–966 (2015)

    MathSciNet  MATH  Google Scholar 

  39. de Laat, T., Vigolo, F.: Superexpanders from group actions on compact manifolds. Geom. Dedicata. arXiv:1707.01399 (2017) (to appear)

  40. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)

    MathSciNet  MATH  Google Scholar 

  41. Doss, S.: Moyennes conditionnelles et martingales dans un espace métrique. C. R. Acad. Sci. Paris 254, 3630–3632 (1962)

    MathSciNet  MATH  Google Scholar 

  42. Dranishnikov, A.N., Gong, G., Lafforgue, V., Yu, G.: Uniform embeddings into Hilbert space and a question of Gromov. Can. Math. Bull. 45(1), 60–70 (2002)

    MathSciNet  MATH  Google Scholar 

  43. Émery, M.: Stochastic Calculus in Manifolds. Universitext. Springer, Berlin (1989). (With an appendix by P.-A. Meyer)

    Google Scholar 

  44. Enflo, P.: On a problem of Smirnov. Ark. Mat. 8, 107–109 (1969)

    MathSciNet  MATH  Google Scholar 

  45. Enflo, P.: On the nonexistence of uniform homeomorphisms between \(L_{p}\)-spaces. Ark. Mat. 8, 103–105 (1969)

    MathSciNet  MATH  Google Scholar 

  46. Enflo, P.: On infinite-dimensional topological groups. In: Séminaire sur la Géométrie des Espaces de Banach (1977–1978), pages Exp. No. 10–11, 11. École Polytech., Palaiseau (1978)

  47. Es-Sahib, A., Heinich, H.: Barycentre canonique pour un espace métrique à courbure négative. In: Séminaire de Probabilités, XXXIII, volume 1709 of Lecture Notes in Mathematics, pp. 355–370. Springer, Berlin (1999)

  48. Eskenazis, A., Mendel, M., Naor, A.: Diamond convexity: a bifurcation in the Ribe program (2018)

  49. Eskenazis, A., Naor, A.: On coarse and uniform embeddings into \({L}_p\). (2018)

  50. Fabian, M., Habala, P., Hájek, P., Montesinos Santalucía, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite-dimensional Geometry, volume 8 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2001)

    MATH  Google Scholar 

  51. Faraut, J., Harzallah, K.: Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier (Grenoble) 24(3), 171–217 (1974)

    MathSciNet  MATH  Google Scholar 

  52. Ferry, S.C., Ranicki, A., Rosenberg, J.: A history and survey of the Novikov conjecture. In: Ferry, S.C., Ranicki, A., Rosenberg, J. (eds.) Novikov Conjectures, Index Theorems and Rigidity, vol. 1 (Oberwolfach, 1993), volume 226 of London Mathematical Society. Lecture Note Series, pp. 7–66. Cambridge University Press, Cambridge (1995)

  53. Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(98), 298–305 (1973)

    MathSciNet  MATH  Google Scholar 

  54. Figiel, T.: On the moduli of convexity and smoothness. Studia Math. 56(2), 121–155 (1976)

    MathSciNet  MATH  Google Scholar 

  55. Fisher, D., Nguyen, T., van Limbeek, W.: Rigidity of warped cones and coarse geometry of expanders. Adv. Math. 346, 665–718 (2019)

    MathSciNet  MATH  Google Scholar 

  56. Foertsch, T., Lytchak, A., Schroeder, V.: Nonpositive curvature and the Ptolemy inequality. Int. Math. Res. Not. IMRN (22):Art. ID rnm100, 15 (2007)

  57. Fujiwara, K., Toyoda, T.: Random groups have fixed points on \(\rm CAT(0)\) cube complexes. Proc. Am. Math. Soc. 140(3), 1023–1031 (2012)

    MathSciNet  MATH  Google Scholar 

  58. Gangolli, R.: Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann. Inst. H. Poincaré Sect. B (N.S.) 3, 121–226 (1967)

    MathSciNet  MATH  Google Scholar 

  59. Garling, D.J.H.: Stable Banach spaces, random measures and Orlicz function spaces. In: Heyer, H. (ed.) Probability Measures on Groups (Oberwolfach, 1981), volume 928 of Lecture Notes in Mathematics, pp. 121–175. Springer, Berlin (1982)

  60. Giladi, O., Mendel, M., Naor, A.: Improved bounds in the metric cotype inequality for Banach spaces. J. Funct. Anal. 260(1), 164–194 (2011)

    MathSciNet  MATH  Google Scholar 

  61. Giladi, O., Naor, A.: Improved bounds in the scaled Enflo type inequality for Banach spaces. Extracta Math. 25(2), 151–164 (2010)

    MathSciNet  MATH  Google Scholar 

  62. Giladi, O., Naor, A., Schechtman, G.: Bourgain’s discretization theorem. Ann. Fac. Sci. Toulouse Math. (6) 21(4), 817–837 (2012)

    MathSciNet  MATH  Google Scholar 

  63. Godefroy, G.: De Grothendieck à Naor: une promenade dans l’analyse métrique des espaces de Banach. Gaz. Math. 151, 13–24 (2017)

    Google Scholar 

  64. Gong, G., Yu, G.: Volume growth and positive scalar curvature. Geom. Funct. Anal. 10(4), 821–828 (2000)

    MathSciNet  MATH  Google Scholar 

  65. Gorin, E.A.: On uniformly topological imbedding of metric spaces in Euclidean and in Hilbert space. Uspehi Mat. Nauk 14(5 (89)), 129–134 (1959)

    MathSciNet  Google Scholar 

  66. Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)

    MathSciNet  MATH  Google Scholar 

  67. Gromov, M.: Rigid transformations groups. In: Géométrie différentielle (Paris. 1986), volume 33 of Travaux en Cours, pp. 65–139. Hermann, Paris (1988)

  68. Gromov, M.: Sign and geometric meaning of curvature. Rend. Sem. Mat. Fis. Milano 61(9–123), 1991 (1994)

    MathSciNet  Google Scholar 

  69. Gromov, M.: Asymptotic invariants of infinite groups. In: Niblo, G.A., Roller, M.A. (eds.) Geometric Group Theory, vol. 2 (Sussex, 1991), volume 182 of London Mathematical Society. Lecture Note Series, pp. 1–295. Cambridge University Press, Cambridge (1993)

  70. Gromov, M.: Spaces and questions. Geom. Funct. Anal. (Special Volume, Part I):118–161 (2000) (GAFA 2000 (Tel Aviv, 1999))

  71. Gromov, M.: \({\rm CAT}(\kappa )\)-spaces: construction and concentration. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 280(Geom. i Topol. 7):100–140, 299–300 (2001)

  72. Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13(1), 73–146 (2003)

    MathSciNet  MATH  Google Scholar 

  73. Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, english edition (2007) (Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates)

  74. Guentner, E., Higson, N., Weinberger, S.: The Novikov conjecture for linear groups. Publ. Math. Inst. Hautes Études Sci. 101, 243–268 (2005)

    MathSciNet  MATH  Google Scholar 

  75. Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)

    MATH  Google Scholar 

  76. Heinrich, S., Mankiewicz, P.: Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Studia Math. 73(3), 225–251 (1982)

    MathSciNet  MATH  Google Scholar 

  77. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43(4), 439–561 (2006)

    MathSciNet  MATH  Google Scholar 

  78. Hrushovski, E.: Extending partial isomorphisms of graphs. Combinatorica 12(4), 411–416 (1992)

    MathSciNet  MATH  Google Scholar 

  79. Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces, vol. II, volume 67 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Cham (2017) (Probabilistic methods and operator theory)

  80. Isbell, J.R.: Uniform Spaces. Mathematical Surveys, No. 12. American Mathematical Society, Providence, RI (1964)

    Google Scholar 

  81. Izeki, H., Nayatani, S.: Combinatorial harmonic maps and discrete-group actions on Hadamard spaces. Geom. Dedicata 114, 147–188 (2005)

    MathSciNet  MATH  Google Scholar 

  82. James, R.C.: Bases and reflexivity of Banach spaces. Ann. Math. 2(52), 518–527 (1950)

    MathSciNet  MATH  Google Scholar 

  83. James, R.C.: A non-reflexive Banach space isometric with its second conjugate space. Proc. Natl. Acad. Sci. U.S.A. 37, 174–177 (1951)

    MathSciNet  MATH  Google Scholar 

  84. James, R.C.: A nonreflexive Banach space that is uniformly nonoctahedral. Isr. J. Math. 18, 145–155 (1974)

    MathSciNet  MATH  Google Scholar 

  85. James, R.C.: Nonreflexive spaces of type \(2\). Isr. J. Math. 30(1–2), 1–13 (1978)

    MathSciNet  MATH  Google Scholar 

  86. James, R.C., Lindenstrauss, J.: The octahedral problem for Banach spaces. Various Publication Series, No. 24, pp. 100–120 (1975)

  87. Johnson, W.B., Randrianarivony, N.L.: \(l_p (p>2)\) does not coarsely embed into a Hilbert space. Proc. Am. Math. Soc. 134(4), 1045–1050 (2006)

    MATH  Google Scholar 

  88. Kalton, N.J.: Coarse and uniform embeddings into reflexive spaces. Q. J. Math. 58(3), 393–414 (2007)

    MathSciNet  MATH  Google Scholar 

  89. Kalton, N.J.: The nonlinear geometry of Banach spaces. Rev. Mat. Complut. 21(1), 7–60 (2008)

    MathSciNet  MATH  Google Scholar 

  90. Kalton, N.J.: Lipschitz and uniform embeddings into \(\ell _\infty \). Fund. Math. 212(1), 53–69 (2011)

    MathSciNet  Google Scholar 

  91. Kalton, N.J., Randrianarivony, N.L.: The coarse Lipschitz geometry of \(l_p\oplus l_q\). Math. Ann. 341(1), 223–237 (2008)

    MathSciNet  MATH  Google Scholar 

  92. Kasparov, G., Yu, G.: The coarse geometric Novikov conjecture and uniform convexity. Adv. Math. 206(1), 1–56 (2006)

    MathSciNet  MATH  Google Scholar 

  93. Kelleher, C., Miller, D., Osborn, T., Weston, A.: Strongly non-embeddable metric spaces. Topol. Appl. 159(3), 749–755 (2012)

    MathSciNet  MATH  Google Scholar 

  94. Khot, S., Naor, A.: Nonembeddability theorems via Fourier analysis. Math. Ann. 334(4), 821–852 (2006)

    MathSciNet  MATH  Google Scholar 

  95. Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. 86, 115–197 (1997)

    MathSciNet  MATH  Google Scholar 

  96. Kondo, T.: \({\rm CAT}(0)\) spaces and expanders. Math. Z. 271(1–2), 343–355 (2012)

    MathSciNet  MATH  Google Scholar 

  97. Krivine, J.-L., Maurey, B.: Espaces de Banach stables. Isr. J. Math. 39(4), 273–295 (1981)

    MathSciNet  MATH  Google Scholar 

  98. Kwapień, S., Schütt, C.: Some combinatorial and probabilistic inequalities and their application to Banach space theory. Studia Math. 82(1), 91–106 (1985)

    MathSciNet  MATH  Google Scholar 

  99. Kwapień, S., Schütt, C.: Some combinatorial and probabilistic inequalities and their application to Banach space theory. II. Studia Math. 95(2), 141–154 (1989)

    MathSciNet  MATH  Google Scholar 

  100. Lafforgue, V.: Un renforcement de la propriété (T). Duke Math. J. 143(3), 559–602 (2008)

    MathSciNet  MATH  Google Scholar 

  101. Lafforgue, V.: Propriété (T) renforcée banachique et transformation de Fourier rapide. J. Topol. Anal. 1(3), 191–206 (2009)

    MathSciNet  MATH  Google Scholar 

  102. Lafforgue, V., Naor, A.: Vertical versus horizontal Poincaré inequalities on the Heisenberg group. Isr. J. Math. 203(1), 309–339 (2014)

    MATH  Google Scholar 

  103. Lafont, J.-F., Prassidis, S.: Roundness properties of groups. Geom. Dedicata 117, 137–160 (2006)

    MathSciNet  MATH  Google Scholar 

  104. Lancien, G., Petitjean, C., Procházka, A.: On the coarse geometry of the James space. arXiv:1805.05171 (2018)

  105. Lang, U., Pavlović, B., Schroeder, V.: Extensions of Lipschitz maps into Hadamard spaces. Geom. Funct. Anal. 10(6), 1527–1553 (2000)

    MathSciNet  MATH  Google Scholar 

  106. Lebedeva, N., Petrunin, A.: Curvature bounded below: a definition a la Berg–Nikolaev. Electron. Res. Announc. Math. Sci. 17, 122–124 (2010)

    MathSciNet  MATH  Google Scholar 

  107. Lee, J.R., Naor, A.: Embedding the diamond graph in \(L_p\) and dimension reduction in \(L_1\). Geom. Funct. Anal. 14(4), 745–747 (2004)

    MathSciNet  MATH  Google Scholar 

  108. Lee, J.R., Naor, A., Peres, Y.: Trees and Markov convexity. Geom. Funct. Anal. 18(5), 1609–1659 (2009)

    MathSciNet  MATH  Google Scholar 

  109. Lennard, C.J., Tonge, A.M., Weston, A.: Generalized roundness and negative type. Mich. Math. J. 44(1), 37–45 (1997)

    MathSciNet  MATH  Google Scholar 

  110. Li, S.: Markov convexity and nonembeddability of the Heisenberg group. Ann. Inst. Fourier (Grenoble) 66(4), 1615–1651 (2016)

    MathSciNet  MATH  Google Scholar 

  111. Li, S., Naor, A.: Discretization and affine approximation in high dimensions. Isr. J. Math. 197(1), 107–129 (2013)

    MathSciNet  MATH  Google Scholar 

  112. Liao, B.: Strong Banach property (T) for simple algebraic groups of higher rank. J. Topol. Anal. 6(1), 75–105 (2014)

    MathSciNet  MATH  Google Scholar 

  113. Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995)

    MathSciNet  MATH  Google Scholar 

  114. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2) 169(3), 903–991 (2009)

    MathSciNet  MATH  Google Scholar 

  115. Matoušek, J.: On embedding expanders into \(l_p\) spaces. Isr. J. Math. 102, 189–197 (1997)

    MATH  Google Scholar 

  116. Maurey, B.: Type, cotype and \(K\)-convexity. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1299–1332. North-Holland, Amsterdam (2003)

  117. Maurey, B., Milman, V.D., Tomczak-Jaegermann, N.: Asymptotic infinite-dimensional theory of Banach spaces. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric aspects of functional analysis (Israel, 1992–1994), volume 77 of Operator Theory: Advances and Applications, pp. 149–175. Birkhäuser, Basel (1995)

    Google Scholar 

  118. Maurey, B., Pisier, G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Math. 58(1), 45–90 (1976)

    MathSciNet  MATH  Google Scholar 

  119. Mendel, M., Naor, A.: Some applications of Ball’s extension theorem. Proc. Am. Math. Soc. 134(9), 2577–2584 (2006)

    MathSciNet  MATH  Google Scholar 

  120. Mendel, M., Naor, A.: Scaled Enflo type is equivalent to Rademacher type. Bull. Lond. Math. Soc. 39(3), 493–498 (2007)

    MathSciNet  MATH  Google Scholar 

  121. Mendel, M., Naor, A.: Metric cotype. Ann. Math. (2) 168(1), 247–298 (2008)

    MathSciNet  MATH  Google Scholar 

  122. Mendel, M., Naor, A.: Towards a calculus for non-linear spectral gaps. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, 17–19 Jan 2010, pp. 236–255 (2010)

  123. Mendel, M., Naor, A.: Spectral calculus and Lipschitz extension for barycentric metric spaces. Anal. Geom. Metr. Spaces 1, 163–199 (2013)

    MathSciNet  MATH  Google Scholar 

  124. Mendel, M., Naor, A.: Nonlinear spectral calculus and super-expanders. Publ. Math. Inst. Hautes Études Sci. 119, 1–95 (2014)

    MathSciNet  MATH  Google Scholar 

  125. Mendel, M., Naor, A.: Expanders with respect to Hadamard spaces and random graphs. Duke Math. J. 164(8), 1471–1548 (2015)

    MathSciNet  MATH  Google Scholar 

  126. Mimura, M.: Sphere equivalence, Banach expanders, and extrapolation. Int. Math. Res. Not. IMRN 12, 4372–4391 (2015)

    MathSciNet  MATH  Google Scholar 

  127. Naor, A.: An application of metric cotype to quasisymmetric embeddings. In: Dai, X., Rong, X. (eds.) Metric and Differential Geometry, volume 297 of Progress in Mathematics, pp. 175–178. Birkhäuser/Springer, Basel (2012)

    Google Scholar 

  128. Naor, A.: An introduction to the Ribe program. Jpn. J. Math. 7(2), 167–233 (2012)

    MathSciNet  MATH  Google Scholar 

  129. Naor, A.: Comparison of metric spectral gaps. Anal. Geom. Metr. Spaces 2, 1–52 (2014)

    MathSciNet  MATH  Google Scholar 

  130. Naor, A.: Discrete Riesz transforms and sharp metric \(X_p\) inequalities. Ann. Math. (2) 184(3), 991–1016 (2016)

    MathSciNet  MATH  Google Scholar 

  131. Naor, A.: A spectral gap precludes low-dimensional embeddings. In: 33rd International Symposium on Computational Geometry, volume 77 of LIPIcs. Leibniz International Proceedings in Informatics, Art. No. 50, 16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2017)

  132. Naor, A.: Metric dimension reduction: a snapshot of the Ribe program. In: Proceedings of the 2018 International Congress of Mathematicians, Rio de Janeiro, vol. I, pp. 759–838 (2018)

  133. Naor, A., Rabani, Y.: On approximate nearest neighbor search in \(\ell _p\) , \(p > 2\). Unpublished manuscript, available on request (2006)

  134. Naor, A., Rabani, Y.: On Lipschitz extension from finite subsets. Isr. J. Math. 219(1), 115–161 (2017)

    MathSciNet  MATH  Google Scholar 

  135. Naor, A., Schechtman, G.: Metric \(X_p\) inequalities. Forum Math. Pi 4(e3), 81 (2016)

    MATH  Google Scholar 

  136. Naor, A., Schechtman, G.: Pythagorean powers of hypercubes. Ann. Inst. Fourier (Grenoble) 66(3), 1093–1116 (2016)

    MathSciNet  MATH  Google Scholar 

  137. Naor, A., Schechtman, G.: Obstructions to metric embeddings of Schatten classes. Preprint (2018)

  138. Naor, A., Silberman, L.: Poincaré inequalities, embeddings, and wild groups. Compos. Math. 147(5), 1546–1572 (2011)

    MathSciNet  MATH  Google Scholar 

  139. Naor, A., Young, R.: Vertical perimeter versus horizontal perimeter. Ann. Math. (2) 188(1), 171–279 (2018)

    MathSciNet  MATH  Google Scholar 

  140. Nowak, P.W., Yu, G.: Large Scale Geometry. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich (2012)

    Google Scholar 

  141. Ohta, S.-I.: Markov type of Alexandrov spaces of non-negative curvature. Mathematika 55(1–2), 177–189 (2009)

    MathSciNet  MATH  Google Scholar 

  142. Osajda, D.: Small cancellation labellings of some infinite graphs and applications. arXiv:1406.5015 (2014)

  143. Ostrovskii, M.I.: Metric Embeddings, volume 49 of De Gruyter Studies in Mathematics. De Gruyter, Berlin (2013). (Bilipschitz and coarse embeddings into Banach spaces)

    Google Scholar 

  144. Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)

    MathSciNet  MATH  Google Scholar 

  145. Ozawa, N.: A note on non-amenability of \({\cal{B}}(l_p)\) for \(p=1,2\). Int. J. Math. 15(6), 557–565 (2004)

    MATH  Google Scholar 

  146. Perelman, G.: Spaces with curvature bounded below. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), pp. 517–525. Birkhäuser, Basel (1995)

  147. Pestov, V.G.: A theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space. Topol. Appl. 155(14), 1561–1575 (2008)

    MathSciNet  MATH  Google Scholar 

  148. Pisier, G.: Martingales with values in uniformly convex spaces. Isr. J. Math. 20(3–4), 326–350 (1975)

    MathSciNet  MATH  Google Scholar 

  149. Pisier, G.: Complex interpolation between Hilbert, Banach and operator spaces. Mem. Am. Math. Soc. 208(978), vi+78 (2010)

    MathSciNet  MATH  Google Scholar 

  150. Pisier, G.: Martingales in Banach Spaces, volume 155 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  151. Pisier, G., Xu, Q.H.: Random series in the real interpolation spaces between the spaces \(v_p\). In: Lindenstrauss, J., Milman, V.D. (eds.) Geometrical Aspects of Functional Analysis (1985/86), volume 1267 of Lecture Notes in Mathematics, pp. 185–209. Springer, Berlin (1987)

  152. Plaut, C.: Metric spaces of curvature \(\ge k\). In: Sher, R.B., Daverman, R.J. (eds.) Handbook of Geometric Topology, pp. 819–898. North-Holland, Amsterdam (2002)

    Google Scholar 

  153. Prassidis, S., Weston, A.: Manifestations of nonlinear roundness in analysis, discrete geometry and topology. In: Arzhantseva, G., Valette, V. (eds.) Limits of Graphs in Group Theory and Computer Science, pp. 141–170. EPFL Press, Lausanne (2009)

    Google Scholar 

  154. Randrianarivony, N.L.: Characterization of quasi-Banach spaces which coarsely embed into a Hilbert space. Proc. Am. Math. Soc. 134(5), 1315–1317 (2006)

    MathSciNet  MATH  Google Scholar 

  155. Raynaud, Y.: Espaces de Banach superstables, distances stables et homéomorphismes uniformes. Isr. J. Math. 44(1), 33–52 (1983)

    MathSciNet  MATH  Google Scholar 

  156. Reingold, O., Vadhan, S., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. Math. (2) 155(1), 157–187 (2002)

    MathSciNet  MATH  Google Scholar 

  157. Ribe, M.: On uniformly homeomorphic normed spaces. Ark. Mat. 14(2), 237–244 (1976)

    MathSciNet  MATH  Google Scholar 

  158. Robertson, G.: Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8(1), 163–172 (1998)

    MathSciNet  MATH  Google Scholar 

  159. Roe, J.: Lectures on Coarse Geometry. volume 31 of University Lecture Series. American Mathematical Society, Providence, RI (2003)

    MATH  Google Scholar 

  160. Roe, J.: Warped cones and property A. Geom. Topol. 9, 163–178 (2005)

    MathSciNet  MATH  Google Scholar 

  161. Rogers, C.A.: A note on coverings and packings. J. Lond. Math. Soc. 25, 327–331 (1950)

    MathSciNet  MATH  Google Scholar 

  162. Sato, T.: An alternative proof of Berg and Nikolaev’s characterization of \({{\rm CAT}}(0)\)-spaces via quadrilateral inequality. Arch. Math. (Basel) 93(5), 487–490 (2009)

    MathSciNet  MATH  Google Scholar 

  163. Sawicki, D.: Super-expanders and warped cones. arXiv:1704.03865 (2017)

  164. Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44(3), 522–536 (1938)

    MathSciNet  MATH  Google Scholar 

  165. Sela, Z.: Uniform embeddings of hyperbolic groups in Hilbert spaces. Isr. J. Math. 80(1–2), 171–181 (1992)

    MathSciNet  MATH  Google Scholar 

  166. Solecki, S.: Extending partial isometries. Isr. J. Math. 150, 315–331 (2005)

    MathSciNet  MATH  Google Scholar 

  167. Sturm, K.T.: Metric spaces of lower bounded curvature. Expos. Math. 17(1), 35–47 (1999)

    MathSciNet  MATH  Google Scholar 

  168. Sturm, K.-T.: Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. Ann. Probab. 30(3), 1195–1222 (2002)

    MathSciNet  MATH  Google Scholar 

  169. Sturm, K.-T.: Probability measures on metric spaces of nonpositive curvature. In: Auscher, P., Coulhon, T., Grigor’yan, A. (eds.) Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), volume 338 of Contemporary Mathematics, pp. 357–390. American Mathematical Society, Providence, RI (2003)

    Google Scholar 

  170. Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)

    MathSciNet  MATH  Google Scholar 

  171. Toyoda, T.: CAT(0) spaces on which a certain type of singularity is bounded. Kodai Math. J. 33(3), 398–415 (2010)

    MathSciNet  MATH  Google Scholar 

  172. Toyoda, T.: Fixed point property for a \({{\rm CAT}}(0)\) space which admits a proper cocompact group action. Kodai Math. J. 39(1), 129–153 (2016)

    MathSciNet  MATH  Google Scholar 

  173. Tsirelson, B.S.: It is impossible to imbed \(\ell _{p}\) or \(c_{0}\) into an arbitrary Banach space. Funkc. Anal. i Prilož. 8(2), 57–60 (1974)

    Google Scholar 

  174. Tukia, P., Väisälä, J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5(1), 97–114 (1980)

    MathSciNet  MATH  Google Scholar 

  175. Tyson, J.T., Wu, J.-M.: Characterizations of snowflake metric spaces. Ann. Acad. Sci. Fenn. Math. 30(2), 313–336 (2005)

    MathSciNet  MATH  Google Scholar 

  176. Veomett, E., Wildrick, K.: Spaces of small metric cotype. J. Topol. Anal. 2(4), 581–597 (2010)

    MathSciNet  MATH  Google Scholar 

  177. Vershik, A.M.: Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space. Topol. Appl. 155(14), 1618–1626 (2008)

    MathSciNet  MATH  Google Scholar 

  178. Vigolo, F.: Measure expanding actions, expanders and warped cones. Trans. Am. Math. Soc. 371(3), 1951–1979 (2019)

    MathSciNet  MATH  Google Scholar 

  179. Villani, C.: Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2009). (Old and new)

    Google Scholar 

  180. Wang, M.-T.: A fixed point theorem of discrete group actions on Riemannian manifolds. J. Differ. Geom. 50(2), 249–267 (1998)

    MathSciNet  MATH  Google Scholar 

  181. Yu, G.: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139(1), 201–240 (2000)

    MathSciNet  MATH  Google Scholar 

  182. Yu, G.: Hyperbolic groups admit proper affine isometric actions on \(l^p\)-spaces. Geom. Funct. Anal. 15(5), 1144–1151 (2005)

    MathSciNet  MATH  Google Scholar 

  183. Yu, G.: Higher index theory of elliptic operators and geometry of groups. In: International Congress of Mathematicians. Vol. II, pages 1623–1639. Eur. Math. Soc., Zürich (2006)

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  1. Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, USA

    Alexandros Eskenazis & Assaf Naor

  2. Mathematics and Computer Science Department, The Open University of Israel, 1 University Road, P.O. Box 808, 43107, Raanana, Israel

    Manor Mendel

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Alexandros Eskenazis was supported by BSF Grant 2010021 and the Simons Foundation. Manor Mendel was supported by BSF Grant 2010021. Assaf Naor was supported by NSF Grant CCF-1412958, BSF Grant 2010021, the Packard Foundation and the Simons Foundation. This work was carried out under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank, and was completed while Assaf Naor was a member of the Institute for Advanced Study.

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Eskenazis, A., Mendel, M. & Naor, A. Nonpositive curvature is not coarsely universal. Invent. math. 217, 833–886 (2019). https://doi.org/10.1007/s00222-019-00878-1

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