Japanese Journal of Mathematics

, Volume 7, Issue 2, pp 167–233 | Cite as

An introduction to the Ribe program

Special Feature: The 10th Takagi Lectures

Abstract

We survey problems, results, ideas, and recent progress in the Ribe program. The goal of this research program, which is motivated by a classical rigidity theorem of Martin Ribe, is to obtain structural results for metric spaces that are inspired by the local theory of Banach spaces. We also present examples of applications of the Ribe program to several areas, including group theory, theoretical computer science, and probability theory.

Keywords and phrases

metric embeddings local theory of Banach spaces bi-Lipschitz and coarse invariants 

Mathematics Subject Classification (2010)

46B07 46B85 46B80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharoni I., Lindenstrauss J.: Uniform equivalence between Banach spaces. Bull. Amer. Math. Soc. 84, 281–283 (1978)MathSciNetMATHGoogle Scholar
  2. 2.
    Aharoni I., Maurey B., Mityagin B.S.: Uniform embeddings of metric spaces and of Banach spaces into Hilbert spaces. Israel J. Math. 52, 251–265 (1985)MathSciNetMATHGoogle Scholar
  3. 3.
    Alon N., Milman V.D.: λ1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38, 73–88 (1985)MathSciNetMATHGoogle Scholar
  4. 4.
    Anderson R.D.: Hilbert space is homeomorphic to the countable infinite product of lines. Bull. Amer. Math. Soc. 72, 515–519 (1966)MathSciNetMATHGoogle Scholar
  5. 5.
    A. Andoni, O. Neiman and A. Naor, Snowflake universality of Wasserstein spaces, preprint, 2010.Google Scholar
  6. 6.
    S. Arora, J.R. Lee and A. Naor, Euclidean distortion and the sparsest cut, J. Amer. Math. Soc., 21 (2008), 1–21 (electronic).Google Scholar
  7. 7.
    S. Arora, L. Lovász, I. Newman, Y. Rabani, Y. Rabinovich and S. Vempala, Local versus global properties of metric spaces (extended abstract), In: Proceedings of the Seventeenth Annual ACM–SIAM Symposium on Discrete Algorithms, ACM, New York, NY, 2006, pp. 41–50.Google Scholar
  8. 8.
    Arzhantseva G., Druţu C., Sapir M.: Compression functions of uniform embeddings of groups into Hilbert and Banach spaces. J. Reine Angew. Math. 633, 213–235 (2009)MathSciNetMATHGoogle Scholar
  9. 9.
    Assouad P.: Plongements lipschitziens dans R n. Bull. Soc. Math. France 111, 429–448 (1983)MathSciNetMATHGoogle Scholar
  10. 10.
    Y. Aumann and Y. Rabani, An O(log k) approximate min-cut max-flow theorem and approximation algorithm, SIAM J. Comput., 27 (1998), 291–301 (electronic).Google Scholar
  11. 11.
    Austin T.: Amenable groups with very poor compression into Lebesgue spaces. Duke Math. J. 159, 187–222 (2011)MathSciNetMATHGoogle Scholar
  12. 12.
    T. Austin and A. Naor, On the bi-Lipschitz structure of Wasserstein spaces, preprint, 2009.Google Scholar
  13. 13.
    Austin T., Naor A., Peres Y.: The wreath product of \({\mathbb{Z}}\) with \({\mathbb{Z}}\) has Hilbert compression exponent \({\frac{2}{3}}\). Proc. Amer. Math. Soc. 137, 85–90 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Austin T., Naor A., Valette A.: The Euclidean distortion of the lamplighter group. Discrete Comput. Geom. 44, 55–74 (2010)MathSciNetMATHGoogle Scholar
  15. 15.
    Ball K.: Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal. 2, 137–172 (1992)MathSciNetMATHGoogle Scholar
  16. 16.
    K. Ball, The Ribe Programme, Séminaire Bourbaki, 1047, 2012.Google Scholar
  17. 17.
    Ball K., Carlen E.A., Lieb E.H.: Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, 463–482 (1994)MathSciNetMATHGoogle Scholar
  18. 18.
    S. Banach, Théorie des Opérations Linéaires, Monografie Matematyczne, 1, PWN—Polish Scientific Publishers, Warsaw, 1932. [Mathematical Monographs, 1].Google Scholar
  19. 19.
    Y. Bartal, On approximating arbitrary metrices by tree metrics, In: STOC ’98, Dallas, TX, ACM, New York, NY, 1999, pp. 161–168.Google Scholar
  20. 20.
    Bartal Y., Bollobás B., Mendel M.: Ramsey-type theorems for metric spaces with applications to online problems. J. Comput. System Sci. 72, 890–921 (2006)MathSciNetMATHGoogle Scholar
  21. 21.
    Bartal Y., Linial N., Mendel M., Naor A.: On metric Ramsey-type dichotomies. J. London Math. Soc. (2) 71, 289–303 (2005)MathSciNetMATHGoogle Scholar
  22. 22.
    Bartal Y., Linial N., Mendel M., Naor A.: On metric Ramsey-type phenomena. Ann. of Math. (2) 162, 643–709 (2005)MathSciNetMATHGoogle Scholar
  23. 23.
    Bartal Y., Linial N., Mendel M., Naor A.: Some low distortion metric Ramsey problems. Discrete Comput. Geom. 33, 27–41 (2005)MathSciNetMATHGoogle Scholar
  24. 24.
    M.A. Bender and M. Farach-Colton, The LCA problem revisited, In: LATIN 2000: Theoretical Informatics, (eds. G.H. Gonnet, D. Panario and A. Viola), Lecture Notes in Comput. Sci., 1776, Springer-Verlag, 2000, pp. 88–94.Google Scholar
  25. 25.
    Y. Benyamini, The uniform classification of Banach spaces, In: Texas Functional Analysis Seminar 1984–1985, Austin, TX, Longhorn Notes, Univ. Texas Press, Austin, TX, 1985, pp. 15–38.Google Scholar
  26. 26.
    Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, Amer. Math. Soc. Colloq. Publ., 48, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  27. 27.
    Bessaga C.: On topological classification of complete linear metric spaces. Fund. Math. 56, 251–288 (1964)MathSciNetGoogle Scholar
  28. 28.
    Bessaga C., Pełczyński A.: Some remarks on homeomorphisms of Banach spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8, 757–761 (1960)Google Scholar
  29. 29.
    C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, 58, PWN—Polish Scientific Publishers, Warsaw, 1975. [Mathematical Monographs, 58].Google Scholar
  30. 30.
    A. Blum, H. Karloff, Y. Rabani and M. Saks, A decomposition theorem for task systems and bounds for randomized server problems, SIAM J. Comput., 30 (2000), 1624–1661 (electronic).Google Scholar
  31. 31.
    Bourgain J.: On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math. 52, 46–52 (1985)MathSciNetMATHGoogle Scholar
  32. 32.
    Bourgain J.: The metrical interpretation of superreflexivity in Banach spaces. Israel J. Math. 56, 222–230 (1986)MathSciNetMATHGoogle Scholar
  33. 33.
    J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, In: Geometrical Aspects of Functional Analysis (1985/86), Lecture Notes in Math., 1267, Springer-Verlag, 1987, pp. 157–167.Google Scholar
  34. 34.
    Bourgain J., Figiel T., Milman V.D.: On Hilbertian subsets of finite metric spaces. Israel J. Math. 55, 147–152 (1986)MathSciNetMATHGoogle Scholar
  35. 35.
    Bourgain J., Milman V.D., Wolfson H.: On type of metric spaces. Trans. Amer. Math. Soc. 294, 295–317 (1986)MathSciNetMATHGoogle Scholar
  36. 36.
    M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, 1999.Google Scholar
  37. 37.
    B. Brinkman, A. Karagiozova and J.R. Lee, Vertex cuts, random walks, and dimension reduction in series-parallel graphs, In: STOC ’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, NY, 2007, pp. 621–630.Google Scholar
  38. 38.
    M. Charikar and A. Karagiozova, A tight threshold for metric Ramsey phenomena, In: Proceedings of the Sixteenth Annual ACM–SIAM Symposium on Discrete Algorithms, ACM, New York, NY, 2005, pp. 129–136 (electronic).Google Scholar
  39. 39.
    M. Charikar, K. Makarychev and Y. Makarychev, Local global tradeoffs in metric embeddings, In: 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), IEEE Computer Soc., Los Alamitos, CA, 2007, pp. 713–723.Google Scholar
  40. 40.
    J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, In: Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969), Princeton Univ. Press, Princeton, NJ, 1970, pp. 195–199.Google Scholar
  41. 41.
    Cheeger J., Kleiner B.: Differentiating maps into L 1, and the geometry of BV functions. Ann. of Math. (2) 171, 1347–1385 (2010)MathSciNetMATHGoogle Scholar
  42. 42.
    P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, Groups with the Haagerup Property. Gromov’s a-T-menability, Progr. Math., 197, Birkhäuser Verlag, Basel, 2001.Google Scholar
  43. 43.
    Christ M., (1990) A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61, 601–628Google Scholar
  44. 44.
    H. Corson and V. Klee, Topological classification of convex sets, In: Proc. Sympos. Pure Math., 7, Amer. Math. Soc., Providence, RI, 1963, pp. 37–51.Google Scholar
  45. 45.
    de Cornulier Y., Tessera R., Valette A.: Isometric group actions on Hilbert spaces: growth of cocycles. Geom. Funct. Anal. 17, 770–792 (2007)MathSciNetMATHGoogle Scholar
  46. 46.
    P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, IL, 2000.Google Scholar
  47. 47.
    A. Dvoretzky, Some results on convex bodies and Banach spaces, In: Proc. Internat. Sympos. Linear Spaces, Jerusalem, 1960, Jerusalem Academic Press, Jerusalem, 1961, pp. 123–160.Google Scholar
  48. 48.
    Enflo P.: On a problem of Smirnov. Ark. Mat. 8, 107–109 (1969)MathSciNetGoogle Scholar
  49. 49.
    Enflo P.: On the nonexistence of uniform homeomorphisms between L p-spaces. Ark. Mat. 8, 103–105 (1969)MathSciNetGoogle Scholar
  50. 50.
    P. Enflo, Uniform structures and square roots in topological groups. I, II, Israel J. Math., 8 (1970), 230–252; ibid., 8 (1970), 253–272.Google Scholar
  51. 51.
    Enflo P.: Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math. 13, 281–288 (1972)MathSciNetGoogle Scholar
  52. 52.
    P. Enflo, Uniform homeomorphisms between Banach spaces, In: Séminaire Maurey–Schwartz (1975–1976), Espaces, L p, Applications Radonifiantes et Géométriedes Espaces de Banach, Exp. No. 18, Centre Math., École Polytech., Palaiseau, 1976, 7 pp.Google Scholar
  53. 53.
    P. Erdős, Extremal problems in graph theory, In: Theory of Graphs and its Applications, Proc. Sympos. Smolenice, 1963, Publ. House Czechoslovak Acad. Sci., Prague, 1964, pp. 29–36.Google Scholar
  54. 54.
    A.G. Èrschler, On the asymptotics of the rate of departure to infinity, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 6, 283 (2001), 251–257, 263.Google Scholar
  55. 55.
    M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books Math./Ouvrages Math. SMC, 8, Springer-Verlag, 2001.Google Scholar
  56. 56.
    Fakcharoenphol J., Rao S., Talwar K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. System Sci. 69, 485–497 (2004)MathSciNetMATHGoogle Scholar
  57. 57.
    X. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, In: École d’Été de Probabilités de Saint-Flour, IV-1974, Lecture Notes in Math., 480, Springer-Verlag, 1975, pp. 1–96.Google Scholar
  58. 58.
    X. Fernique, Évaluations de processus gaussiens composés, In: Probability in Banach spaces, Proc. First Internat. Conf., Oberwolfach, 1975, Lecture Notes in Math., 526, Springer-Verlag, 1976, pp. 67–83.Google Scholar
  59. 59.
    X. Fernique, Caractérisation de processus à trajectoires majorées ou continues, In: Séminaire de Probabilités, XII, Univ. Strasbourg, Strasbourg, 1976/1977, Lecture Notes in Math., 649, Springer-Verlag, 1978, pp. 691–706.Google Scholar
  60. 60.
    Figiel T.: On nonlinear isometric embeddings of normed linear spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16, 185–188 (1968)MathSciNetMATHGoogle Scholar
  61. 61.
    Figiel T., Lindenstrauss J., Milman V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. 139, 53–94 (1977)MathSciNetMATHGoogle Scholar
  62. 62.
    M. Fréchet, Les Espaces Abstraits et leur Théorie Considérée comme Introduction à L’analyse Générale, Collection de Monographies sur la Théorie des Fonctions, Gauthier-Villars, Paris, 12, 1928, 296 pp.Google Scholar
  63. 63.
    Giladi O., Mendel M., Naor A.: Improved bounds in the metric cotype inequality for Banach spaces. J. Funct. Anal. 260, 164–194 (2011)MathSciNetMATHGoogle Scholar
  64. 64.
    O. Giladi, A. Naor and G. Schechtman, Bourgain’s discretization theorem, preprint, arXiv:1110.5368; Ann. Fac. Sci. Toulouse Math., to appear (2011).Google Scholar
  65. 65.
    Gorelik E.: The uniform nonequivalence of L p and l p. Israel J. Math. 87, 1–8 (1994)MathSciNetMATHGoogle Scholar
  66. 66.
    Gromov M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982)MathSciNetMATHGoogle Scholar
  67. 67.
    Gromov M.: Filling Riemannian manifolds. J. Differential Geom. 18, 1–147 (1983)MathSciNetMATHGoogle Scholar
  68. 68.
    M. Gromov, Asymptotic invariants of infinite groups, In: Geometric Group Theory. Vol. 2, Sussex, 1991, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295.Google Scholar
  69. 69.
    M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates, Mod. Birkhäuser Class., Birkhäuser Boston Inc., Boston, MA, English ed., 2007.Google Scholar
  70. 70.
    Grothendieck A.: Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky–Rogers. Bol. Soc. Mat. São Paulo 8, 81–110 (1953)MathSciNetGoogle Scholar
  71. 71.
    A. Grothendieck, Erratum au mémoire: Produits tensoriels topologiques et espaces nucléaires, Ann. Inst. Fourier (Grenoble), 6 (1955–1956), 117–120.Google Scholar
  72. 72.
    Guentner E., Kaminker J.: Exactness and uniform embeddability of discrete groups. J. London Math. Soc. (2) 70, 703–718 (2004)MathSciNetMATHGoogle Scholar
  73. 73.
    Gupta A., Newman I., Rabinovich Y., Sinclair A.: Cuts, trees and l 1-embeddings of graphs. Combinatorica 24, 233–269 (2004)MathSciNetMATHGoogle Scholar
  74. 74.
    Hanner O.: On the uniform convexity of L p and l p. Ark. Mat. 3, 239–244 (1956)MathSciNetMATHGoogle Scholar
  75. 75.
    S. Har-Peled and M. Mendel, Fast construction of nets in low-dimensional metrics and their applications, SIAM J. Comput., 35 (2006), 1148–1184 (electronic).Google Scholar
  76. 76.
    Harel D., Tarjan R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13, 338–355 (1984)MathSciNetMATHGoogle Scholar
  77. 77.
    Heinrich S., Mankiewicz P.: Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Studia Math. 73, 225–251 (1982)MathSciNetMATHGoogle Scholar
  78. 78.
    Henkin G.M.: The lack of a uniform homeomorphism between the spaces of smooth functions of one and of n variables \({(n \geq 2)}\). Mat. Sb. (N.S.) 74, 595–607 (1967)MathSciNetGoogle Scholar
  79. 79.
    S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.), 43 (2006), 439–561 (electronic).Google Scholar
  80. 80.
    Howroyd J.D.: On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. London Math. Soc. (3) 70, 581–604 (1995)MathSciNetMATHGoogle Scholar
  81. 81.
    Hughes B.: Trees and ultrametric spaces: a categorical equivalence. Adv. Math. 189, 148–191 (2004)MathSciNetMATHGoogle Scholar
  82. 82.
    T. Hytönen, S. Li and A. Naor, Quantitative affine approximation for UMD targets, preprint, 2012.Google Scholar
  83. 83.
    T. Hytönen and A. Naor, Pisier’s inequality revisited, preprint, 2012.Google Scholar
  84. 84.
    R.C. James, Some self-dual properties of normed linear spaces, In: Symposium on Infinite-Dimensional Topology, Louisiana State Univ., Baton Rouge, LA, 1967, Ann. of Math. Stud., 69, Princeton Univ. Press, Princeton, NJ, 1972, pp. 159–175.Google Scholar
  85. 85.
    James R.C.: Super-reflexive Banach spaces. Canad. J. Math. 24, 896–904 (1972)MathSciNetMATHGoogle Scholar
  86. 86.
    James R.C.: Nonreflexive spaces of type 2. Israel J. Math. 30, 1–13 (1978)MathSciNetMATHGoogle Scholar
  87. 87.
    F. John, Extremum problems with inequalities as subsidiary conditions, In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, NY, 1948, pp. 187–204.Google Scholar
  88. 88.
    W.B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, In: Conference in Modern Analysis and Probability, New Haven, CN, 1982, Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984, pp. 189–206.Google Scholar
  89. 89.
    Johnson W.B., Lindenstrauss J., Schechtman G.: Banach spaces determined by their uniform structures. Geom. Funct. Anal. 6, 430–470 (1996)MathSciNetMATHGoogle Scholar
  90. 90.
    Johnson W.B., Schechtman G.: Diamond graphs and super-reflexivity. J. Topol. Anal. 1, 177–189 (2009)MathSciNetMATHGoogle Scholar
  91. 91.
    M.Ĭ. Kadec′, On strong and weak convergence, Dokl. Akad. Nauk SSSR, 122 (1958), 13–16.Google Scholar
  92. 92.
    M.Ĭ. Kadec′, Topological equivalence of all separable Banach spaces, Dokl. Akad. Nauk SSSR, 167 (1966), 23–25.Google Scholar
  93. 93.
    M.Ĭ. Kadec′, A proof of the topological equivalence of all separable infinite-dimensional Banach spaces, Funkcional. Anal. i Priložen., 1 (1967), 61–70.Google Scholar
  94. 94.
    Kahane J.-P.: Sur les sommes vectorielles \({\sum \pm u_{n}}\). C. R. Acad. Sci. Paris 259, 2577– (1964)MathSciNetMATHGoogle Scholar
  95. 95.
    N.J. Kalton, The uniform structure of Banach spaces, Math. Ann., to appear (2011).Google Scholar
  96. 96.
    Kalton N.J.: The nonlinear geometry of Banach spaces. Rev. Mat. Complut. 21, 7–60 (2008)MathSciNetMATHGoogle Scholar
  97. 97.
    Karloff H., Rabani Y., Ravid Y.: Lower bounds for randomized k-server and motion-planning algorithms. SIAM J. Comput. 23, 293–312 (1994)MathSciNetMATHGoogle Scholar
  98. 98.
    T. Keleti, A. Máthé and O. Zindulka, Hausdorff dimension of metric spaces and Lipschitz maps onto cubes, preprint, arXiv:1203.0686.Google Scholar
  99. 99.
    Khot S., Naor A.: Nonembeddability theorems via Fourier analysis. Math. Ann. 334, 821–852 (2006)MathSciNetMATHGoogle Scholar
  100. 100.
    S. Khot and R. Saket, SDP integrality gaps with local \({\ell_1}\)-embeddability, In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), IEEE Computer Soc., Los Alamitos, CA, 2009, pp. 565–574.Google Scholar
  101. 101.
    Laakso T.J.: Plane with A -weighted metric not bi-Lipschitz embeddable to \({\mathbb{R}^{N}}\). Bull. London Math. Soc. 34, 667–676 (2002)MathSciNetMATHGoogle Scholar
  102. 102.
    Lang U., Plaut C.: Bilipschitz embeddings of metric spaces into space forms. Geom. Dedicata 87, 285–307 (2001)MathSciNetMATHGoogle Scholar
  103. 103.
    M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergeb. Math. Grenzgeb. (3), 23, Springer-Verlag, 1991.Google Scholar
  104. 104.
    Lee J.R., Naor A., Peres Y.: Trees and Markov convexity. Geom. Funct. Anal. 18, 1609–1659 (2009)MathSciNetMATHGoogle Scholar
  105. 105.
    S. Li and A. Naor, Discretization and affine approximation in high dimensions, preprint, arXiv:1202.2567; Israel J. Math., to appear (2012).Google Scholar
  106. 106.
    Lindenstrauss J.: On the modulus of smoothness and divergent series in Banach spaces. Michigan Math. J. 10, 241–252 (1963)MathSciNetGoogle Scholar
  107. 107.
    Lindenstrauss J.: On nonlinear projections in Banach spaces. Michigan Math. J. 11, 263–287 (1964)MathSciNetMATHGoogle Scholar
  108. 108.
    Lindenstrauss J.: Some aspects of the theory of Banach spaces. Advances in Math. 5, 159–180 (1970)MathSciNetMATHGoogle Scholar
  109. 109.
    J. Lindenstrauss, Uniform embeddings, homeomorphisms and quotient maps between Banach spaces (a short survey), In: 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra, 1996, Topology Appl., 85, Elsevier Sci. B. V., Amsterdam, 1998, pp. 265–279.Google Scholar
  110. 110.
    N. Linial, Finite metric-spaces—combinatorics, geometry and algorithms, In: Proceedings of the International Congress of Mathematicians, 3, Beijing, 2002, Higher Ed. Press, Beijing, 2002, pp. 573–586.Google Scholar
  111. 111.
    Linial N., London E., Rabinovich Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 215–245 (1995)MathSciNetMATHGoogle Scholar
  112. 112.
    Linial N., Magen A., Naor A.: Girth and Euclidean distortion. Geom. Funct. Anal. 12, 380–394 (2002)MathSciNetMATHGoogle Scholar
  113. 113.
    Linial N., Saks M.: The Euclidean distortion of complete binary trees. Discrete Comput. Geom. 29, 19–21 (2003)MathSciNetMATHGoogle Scholar
  114. 114.
    K. Makarychev and Y. Makarychev, Metric extension operators, vertex sparsifiers and Lipschitz extendability, In: 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), IEEE Computer Soc., Los Alamitos, CA, 2010, pp. 255–264.Google Scholar
  115. 115.
    Marcus M.B., Pisier G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152, 245–301 (1984)MathSciNetMATHGoogle Scholar
  116. 116.
    Matoušek J.: Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces. Comment. Math. Univ. Carolin. 33, 451–463 (1992)MathSciNetMATHGoogle Scholar
  117. 117.
    Matoušek J.: On embedding trees into uniformly convex Banach spaces. Israel J. Math. 114, 221–237 (1999)MathSciNetMATHGoogle Scholar
  118. 118.
    J. Matoušek, Lectures on Discrete Geometry, Grad. Texts in Math., 212, Springer-Verlag, 2002.Google Scholar
  119. 119.
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Stud. Adv. Math., 44, Cambridge Univ. Press, Cambridge, 1995.Google Scholar
  120. 120.
    B. Maurey, Théorèmes de Factorisation pour les Opérateurs Linéaires à Valeurs dans les Espaces L p, With an English summary, Astérisque, 11, Soc. Math. France, Paris, 1974.Google Scholar
  121. 121.
    B. Maurey, Type, cotype and K-convexity, In: Handbook of the Geometry of Banach Spaces, 2, North-Holland, Amsterdam, 2003, pp. 1299–1332.Google Scholar
  122. 122.
    Maurey B., Pisier G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Math. 58, 45–90 (1976)MathSciNetMATHGoogle Scholar
  123. 123.
    Mazur S., Ulam S.: Sur les transformations isométriques d’espaces vectoriels normés. C. R. Math. Acad. Sci. Paris 194, 946–948 (1932)Google Scholar
  124. 124.
    M. Mendel, Metric dichotomies, In: Limits of Graphs in Group Theory and Computer Science, EPFL Press, Lausanne, 2009, pp. 59–76.Google Scholar
  125. 125.
    Mendel M., Naor A.: Euclidean quotients of finite metric spaces. Adv. Math. 189, 451–494 (2004)MathSciNetMATHGoogle Scholar
  126. 126.
    M. Mendel and A. Naor, Some applications of Ball’s extension theorem, Proc. Amer. Math. Soc., 134 (2006), 2577–2584 (electronic).Google Scholar
  127. 127.
    Mendel M., Naor A.: Ramsey partitions and proximity data structures. J. Eur. Math. Soc. (JEMS) 9, 253–275 (2007)MathSciNetMATHGoogle Scholar
  128. 128.
    Mendel M., Naor A.: Scaled Enflo type is equivalent to Rademacher type. Bull. Lond. Math. Soc. 39, 493–498 (2007)MathSciNetMATHGoogle Scholar
  129. 129.
    M. Mendel and A. Naor, Markov convexity and local rigidity of distorted metrics [extended abstract], In: Computational Geometry (SCG ’08), ACM, New York, NY, 2008, pp. 49–58; Full version, J. Eur. Math. Soc. (JEMS), to appear.Google Scholar
  130. 130.
    Mendel M., Naor A.: Metric cotype. Ann. of Math. (2) 168, 247–298 (2008)MathSciNetMATHGoogle Scholar
  131. 131.
    Mendel M., Naor A.: Maximum gradient embeddings and monotone clustering. Combinatorica 30, 581–615 (2010)MathSciNetMATHGoogle Scholar
  132. 132.
    M. Mendel and A. Naor, Towards a calculus for non-linear spectral gaps, In: Proceedings of the Twenty-First Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2010, pp. 236–255.Google Scholar
  133. 133.
    A. Andoni, O. Neiman and A. Naor, Snowflake universality of Wasserstein spaces, preprint, 2010.Google Scholar
  134. 134.
    M. Mendel and A. Naor, Ultrametric skeletons, preprint, arXiv:1112.3416; Proc. Natl. Acad. Sci. USA, to appear (2011).Google Scholar
  135. 135.
    M. Mendel and A. Naor, Spectral calculus and Lipschitz extension for barycentric metric spaces, preprint, 2012.Google Scholar
  136. 136.
    M. Mendel and A. Naor, Ultrametric subsets with large Hausdorff dimension, Invent. Math., DOI:10.1007/s00222-012-0402-7, 2012.
  137. 137.
    M. Mendel and C. Schwob, Fast C-K-R partitions of sparse graphs, Chic. J. Theoret. Comput. Sci., 2009 (2009), Article 2, 15 pp.Google Scholar
  138. 138.
    V.D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen., 5 (1971), 28–37.Google Scholar
  139. 139.
    Milman V.D.: Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space. Proc. Amer. Math. Soc. 94, 445–449 (1985)MathSciNetMATHGoogle Scholar
  140. 140.
    V.D. Milman and G. Schechtman, An “isomorphic” version of Dvoretzky’s theorem. II, In: Convex Geometric Analysis, Berkeley, CA, 1996, Math. Sci. Res. Inst. Publ., 34, Cambridge Univ. Press, Cambridge, 1999, pp. 159–164.Google Scholar
  141. 141.
    Milnor J., Thurston W.: Characteristic numbers of 3-manifolds. Enseignement Math. (2) 23, 249–254 (1977)MathSciNetMATHGoogle Scholar
  142. 142.
    P.B. Miltersen, Cell probe complexity—a survey, In: 19th Conference on the Foundations of Software Technology and Theoretical Computer Science, 1999.Google Scholar
  143. 143.
    Mostow G.D.: Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math. 34, 53–104 (1968)MathSciNetMATHGoogle Scholar
  144. 144.
    Naor A.: A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between L p spaces. Mathematika 48, 253–271 (2001)MathSciNetMATHGoogle Scholar
  145. 145.
    A. Naor, An application of metric cotype to quasisymmetric embeddings, preprint, arXiv:math/0607644.Google Scholar
  146. 146.
    A. Naor, L 1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry, In: Proceedings of the International Congress of Mathematicians, 3, Hindustan Book Agency, New Delhi, 2010, pp. 1549–1575.Google Scholar
  147. 147.
    A. Naor and Y. Peres, Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not. IMRN, 2008 (2008), Art. ID rnn 076, 34 pp.Google Scholar
  148. 148.
    Naor A., Peres Y.: L p compression, traveling salesmen, and stable walks. Duke Math. J. 157, 53–108 (2011)MathSciNetMATHGoogle Scholar
  149. 149.
    Naor A., Peres Y., Schramm O., Sheffield S.: Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J. 134, 165–197 (2006)MathSciNetMATHGoogle Scholar
  150. 150.
    Naor A., Schechtman G.: Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math. 552, 213–236 (2002)MathSciNetMATHGoogle Scholar
  151. 151.
    Naor A., Tao T.: Random martingales and localization of maximal inequalities. J. Funct. Anal. 259, 731–779 (2010)MathSciNetMATHGoogle Scholar
  152. 152.
    A. Naor and T. Tao, Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem, preprint, arXiv:1003.4013; Israel J. Math., to appear (2010).Google Scholar
  153. 153.
    Nazarov F., Treil S., Volberg A.: The Tb-theorem on non-homogeneous spaces. Acta Math. 190, 151–239 (2003)MathSciNetMATHGoogle Scholar
  154. 154.
    Ohta S.-I.: Markov type of Alexandrov spaces of non-negative curvature. Mathematika 55, 177–189 (2009)MathSciNetMATHGoogle Scholar
  155. 155.
    A. Olshanskii and D. Osin, A quasi-isometric embedding theorem for groups, preprint, arXiv:1202.6437.Google Scholar
  156. 156.
    Ostrovskii M.I.: On metric characterizations of some classes of Banach spaces. C. R. Acad. Bulgare Sci. 64, 775–784 (2011)MathSciNetMATHGoogle Scholar
  157. 157.
    M.I. Ostrovskii, Different forms of metric characterizations of classes of Banach spaces, preprint, arXiv:1112.0801.Google Scholar
  158. 158.
    Ostrovskii M.I.: Low-distortion embeddings of graphs with large girth. J. Funct. Anal. 262, 3548–3555 (2012)MathSciNetMATHGoogle Scholar
  159. 159.
    M.I. Ostrovskii, Test-space characterizations of some classes of Banach spaces, preprint, arXiv:1112.3086.Google Scholar
  160. 160.
    Pansu P.: Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129(2), 1–60 (1989)MathSciNetMATHGoogle Scholar
  161. 161.
    Pisier G.: Sur les espaces de Banach qui ne contiennent pas uniformément de \({l^{1}_{n}}\). C. R. Acad. Sci. Paris Sér. A-B 277, A991–A994 (1973)MathSciNetGoogle Scholar
  162. 162.
    Pisier G.: with values in uniformly convex spaces. Israel J. Math. 20, 326–350 (1975)MathSciNetMATHGoogle Scholar
  163. 163.
    Pisier G.: Holomorphic semigroups and the geometry of Banach spaces. Ann. of Math. 115(2), 375–392 (1982)MathSciNetMATHGoogle Scholar
  164. 164.
    G. Pisier, Probabilistic methods in the geometry of Banach spaces, In: Probability and Analysis, Varenna, 1985, Lecture Notes in Math., 1206, Springer-Verlag, 1986, pp. 167–241.Google Scholar
  165. 165.
    G. Pisier and Q.H. Xu, Random series in the real interpolation spaces between the spaces v p, In: Geometrical Aspects of Functional Analysis, Israel Seminar, 1985–1986, Lecture Notes in Math., 1267, Springer-Verlag, 1987, pp. 185–209.Google Scholar
  166. 166.
    Rabinovich Y.: On average distortion of embedding metrics into the line. Discrete Comput. Geom. 39, 720–733 (2008)MathSciNetMATHGoogle Scholar
  167. 167.
    P. Raghavendra and D. Steurer, Integrality gaps for strong SDP relaxations of unique games, In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), IEEE Computer Soc., Los Alamitos, CA, 2009, pp. 575–585.Google Scholar
  168. 168.
    Revelle D.: Rate of escape of random walks on wreath products and related groups. Ann. Probab. 31, 1917–1934 (2003)MathSciNetMATHGoogle Scholar
  169. 169.
    Ribe M.: On uniformly homeomorphic normed spaces. Ark. Mat. 14, 237–244 (1976)MathSciNetMATHGoogle Scholar
  170. 170.
    Ribe M.: On uniformly homeomorphic normed spaces. II, Ark. Mat. 16, 1–9 (1978)MathSciNetMATHGoogle Scholar
  171. 171.
    Ribe M.: Existence of separable uniformly homeomorphic nonisomorphic Banach spaces. Israel J. Math. 48, 139–147 (1984)MathSciNetMATHGoogle Scholar
  172. 172.
    Sachs H.: Regular graphs with given girth and restricted circuits. J. London Math. Soc. 38, 423–429 (1963)MathSciNetMATHGoogle Scholar
  173. 173.
    Sagan H.: Space-Filling Curves. Springer-Verlag, Universitext (1994)MATHGoogle Scholar
  174. 174.
    Schechtman G.: Two observations regarding embedding subsets of Euclidean spaces in normed spaces. Adv. Math. 200, 125–135 (2006)MathSciNetMATHGoogle Scholar
  175. 175.
    Semmes S.: On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A -weights. Rev. Mat. Iberoamericana 12, 337–410 (1996)MathSciNetMATHGoogle Scholar
  176. 176.
    Solé P.: The second eigenvalue of regular graphs of given girth. J. Combin. Theory Ser. B 56, 239–249 (1992)MathSciNetGoogle Scholar
  177. 177.
    C. Sommer, E. Verbin and W. Yu, Distance oracles for sparse graphs, In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), IEEE Computer Soc., Los Alamitos, CA, 2009, pp. 703–712.Google Scholar
  178. 178.
    Talagrand M.: Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987)MathSciNetMATHGoogle Scholar
  179. 179.
    Talagrand M.: Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geom. Funct. Anal. 3, 295–314 (1993)MathSciNetMATHGoogle Scholar
  180. 180.
    M. Talagrand The Generic Chaining. Upper and Lower Bounds of Stochastic Processes, Springer Monogr. Math., Springer-Verlag, 2005.Google Scholar
  181. 181.
    M. Talagrand, Upper and Lower Bounds for Stochastic Processes, Modern Methods and Classical Problems, forthcoming book.Google Scholar
  182. 182.
    M. Thorup and U. Zwick, Approximate distance oracles, J. ACM, 52 (2005), 1–24 (electronic).Google Scholar
  183. 183.
    Toruńczyk H.: Characterizing Hilbert space topology. Fund. Math. 111, 247–262 (1981)MathSciNetMATHGoogle Scholar
  184. 184.
    Urbański M.: Transfinite Hausdorff dimension. Topology Appl. 156, 2762–2771 (2009)MathSciNetMATHGoogle Scholar
  185. 185.
    J. Väisälä, The free quasiworld. Freely quasiconformal and related maps in Banach spaces, In: Quasiconformal Geometry and Dynamics, Lublin, 1996, Banach Center Publ., 48, Polish Acad. Sci., Warsaw, 1999, pp. 55–118.Google Scholar
  186. 186.
    Vestfrid I.A., Timan A.F.: A universality property of Hilbert spaces, Dokl. Akad. Nauk SSSR. 246, 528–530 (1979)MathSciNetGoogle Scholar
  187. 187.
    C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., 58, Amer. Math. Soc., Providence, RI, 2003.Google Scholar
  188. 188.
    R. Wagner, Notes on an inequality by Pisier for functions on the discrete cube, In: Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1745, Springer-Verlag, 2000, pp. 263–268.Google Scholar
  189. 189.
    Williamson D.P., Shmoys D.B.: The Design of Approximation Algorithms. Cambridge Univ. Press, Cambridge (2011)MATHGoogle Scholar
  190. 190.
    Wojtaszczyk P., Banach Spaces for Analysts, Cambridge Stud. Adv. Math., 25, Cambridge Univ. Press, Cambridge, 1991.Google Scholar
  191. 191.
    Wulff-Nilsen C. Approximate distance oracles with improved query time, preprint, arXiv:1202.2336.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2012

Authors and Affiliations

  1. 1.Courant Institute, New York UniversityNew YorkUSA

Personalised recommendations