Israel Journal of Mathematics

, Volume 219, Issue 1, pp 115–161 | Cite as

On Lipschitz extension from finite subsets

  • Assaf Naor
  • Yuval Rabani


We prove that for every n ∈ ℕ there exists a metric space (X, d X), an n-point subset SX, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: SZ such that the Lipschitz constant of every function F: XZ that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ℕ there exists a metric space (X, d X), an n-point subset S ⊆ X and a function f: S → ℓ2 that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ℓ2 that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ach03]
    D. Achlioptas, Database-friendly random projections: Johnson–Lindenstrauss with binary coins, Journal of Computer and System Sciences 66 (2003), 671–687.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AFH+04]
    A. Archer, J. Fakcharoenphol, C. Harrelson, R. Krauthgamer, K. Talwar and É. Tardos, Approximate classification via earthmover metrics, in Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2004, pp. 1079–1087 (electronic).Google Scholar
  3. [Bal92]
    K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric and Functional Analysis 2 (1992), 137–172.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Bal13]
    Keith Ball, The Ribe programme, Astérisque 352 (2013), Exp. No. 1047, viii, 147–159.Google Scholar
  5. [Beg99]
    B. Begun, A remark on almost extensions of Lipschitz functions, Israel Journal of Mathematics 109 (1999), 151–155.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BFdlV82]
    B. Bollobás and W. Fernandez de la Vega, The diameter of random regular graphs, Combinatorica 2 (1982), 125–134.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BL00]
    Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications, Vol. 48, American Mathematical Society, Providence, RI, 2000.zbMATHGoogle Scholar
  8. [Bou81]
    J. Bourgain, A counterexample to a complementation problem, Compositio Mathematica 43 (1981), 133–144.MathSciNetzbMATHGoogle Scholar
  9. [Bou87]
    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 Mathematics, Vol. 1267, Springer, Berlin, 1987, pp. 157–167.CrossRefGoogle Scholar
  10. [CKNZ05]
    C. Chekuri, S. Khanna, J. Naor and L. Zosin, A linear programming formulation and approximation algorithms for the metric labeling problem, SIAM Journal on Discrete Mathematics 18 (2004/05), 608–625.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [CKR05]
    G. Calinescu, H. Karloff and Y. Rabani, Approximation algorithms for the 0-extension problem, SIAM Journal on Computing 34 (2004/05), 358–372.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Die10]
    R. Diestel, Graph Theory, Graduate Texts in Mathematics, Vol. 173, Springer, Heidelberg, 2010.CrossRefzbMATHGoogle Scholar
  13. [D.J.P.+94]
    [DJP+94]_E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour and M. Yannakakis, The complexity of multiterminal cuts, SIAM Journal on Computing 23 (1994), 864–894.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [Dvo61]
    A. Dvoretzky, Some results on convex bodies and Banach spaces, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160..Google Scholar
  15. [Enf69]
    P. Enflo, On the nonexistence of uniform homeomorphisms between Lp-spaces, Arkiv för Matematik 8 (1969), 103–105.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [FHRT03]
    J. Fakcharoenphol, C. Harrelson, S. Rao and K. Talwar, An improved approximation algorithm for the 0-extension problem, in Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003), ACM, New York, 2003, pp. 257–265.Google Scholar
  17. [FJS88]
    T. Figiel, W. B. Johnson and G. Schechtman, Factorizations of natural embeddings of ln p into Lr. I, Studia Mathematica 89 (1988), 79–103.MathSciNetzbMATHGoogle Scholar
  18. [FLM77]
    T. Figiel, J. Lindenstrauss and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Mathematica 139 (1977), 53–94.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [FTJ79]
    T. Figiel and N. Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel Journal of Mathematics 33 (1979), 155–171.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [HLW06]
    S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bulletin of the American Mathematical Society 43 (2006), 439–561 (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  21. [HW71]
    T. L. Hayden and J. H. Wells, On the extension of Lipschitz–Hölder maps of order ß, Journal of Mathematical Analysis and Applications 33 (1971), 627–640.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [JL84]
    W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, in Conference in Modern Analysis and Probability (New Haven, Conn., 1982), Contemporary Mathematics, Vol. 26, American Mathematical Society, Providence, RI, 1984, pp. 189–206.CrossRefGoogle Scholar
  23. [JLS86]
    W. B. Johnson, J. Lindenstrauss and G. Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel Journal of Mathematics 54 (1986), 129–138.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Joh48]
    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, New York, 1948, pp. 187–204.Google Scholar
  25. [Kal04]
    N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collectanea Mathematica 55 (2004), 171–217.MathSciNetzbMATHGoogle Scholar
  26. [Kal12]
    N. J. Kalton, The uniform structure of Banach spaces, Mathematische Annalen 354 (2012), 1247–1288.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Kar98]
    A. V. Karzanov, Minimum 0-extensions of graph metrics, European Journal of Combinatorics 19 (1998), 71–101.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Kaš77]
    B. S. Kašin, The widths of certain finite-dimensional sets and classes of smooth functions, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 41 (1977), 334–351, 478.MathSciNetGoogle Scholar
  29. [Kir34]
    M. D. Kirszbraun, Über die zusammenziehenden und Lipschitzchen Transformationen, Fundamenta Mathematicae 22 (1934), 77–108.zbMATHGoogle Scholar
  30. [KKMR09]
    H. Karloff, S. Khot, A. Mehta and Y. Rabani, On earthmover distance, metric labeling, and 0-extension, SIAM Journal on Computing 39 (2009), 371–387.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [KRTJ80]
    H. König, J. R. Retherford and N. Tomczak-Jaegermann, On the eigenvalues of (p, 2)-summing operators and constants associated with normed spaces, Journal of Functional Analysis 37 (1980), 88–126.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [KS71]
    M. Ĭ. Kadec and M. G. Snobar, Certain functionals on the Minkowski compactum, Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki 10 (1971), 453–457.MathSciNetGoogle Scholar
  33. [Lan99]
    U. Lang, Extendability of large-scale Lipschitz maps, Transactions of the American Mathematical Society 351 (1999), 3975–3988.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Lew78]
    D. R. Lewis, Finite dimensional subspaces of Lp, Studia Mathematica 63 (1978), 207–212.MathSciNetzbMATHGoogle Scholar
  35. [Lin64]
    J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Mathematical Journal 11 (1964), 263–287.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [LN03]
    J. R. Lee and A. Naor, Metric decomposition, smooth measures, and clustering, preprint, available on request, 2003.Google Scholar
  37. [LN04]
    J. R. Lee and A. Naor, Absolute Lipschitz extendability Comptes Rendus Mathématique. Académie des Sciences. Paris 338 (2004), 859–862.Google Scholar
  38. [LN05]
    J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones Mathematicae 160 (2005), 59–95.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [Mat97]
    J. Matoušek, On embedding expanders into lp spaces, Israel Journal of Mathematics 102 (1997), 189–197.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [Mat02]
    J. Matoušek, Lectures on Discrete Geometry, Graduate Texts in Mathematics, Vol. 212, Springer-Verlag, New York, 2002.CrossRefGoogle Scholar
  41. [Mau74]
    B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces L p, Société Mathématique de France, Paris, 1974.zbMATHGoogle Scholar
  42. [Men27]
    K. Menger, Zur allgemeinen Kurventheorie, Fundamenta Mathematicae 10 (1927), 96–115.zbMATHGoogle Scholar
  43. [Min70]
    G. J. Minty, On the extension of Lipschitz, Lipschitz–Hölder continuous, and monotone functions, Bulletin of the American Mathematical Society 76 (1979), 334–339.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [MM10]
    K. Makarychev and Y. Makarychev, Metric extension operators, vertex sparsifiers and Lipschitz extendability, in IEEE 51st Annual Symposium on Foundations of Computer Science FOCS 2010, IEEE Computer Soc., Los Alamitos, CA, 2010, pp. 255–264.CrossRefGoogle Scholar
  45. [MN13a]
    M. Mendel and A. Naor, Spectral calculus and Lipschitz extension for barycentric metric spaces, Analysis and Geometry in Metric Spaces 1 (2013), 163–199.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [MN13b]
    M. Mendel and A. Naor, Ultrametric skeletons, Proceedings of the National Academy of Sciences of the United States of America 110 (2013), 19256–19262.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [MN15]
    M. Mendel and A. Naor, Expanders with respect to Hadamard spaces and random graphs, Duke Mathematical Journal 164 (2015), 1471–1548.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [MP84]
    M. B. Marcus and G. Pisier, Characterizations of almost surely continuous pstable random Fourier series and strongly stationary processes, Acta Mathematica 152 (1984), 245–301.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [MS86]
    V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-dimensional Normed Spaces, Lecture Notes in Mathematics, Vol. 1200, Springer-Verlag, Berlin, 1986.Google Scholar
  50. [Nao01]
    A. Naor, A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between Lp spaces, Mathematika 48 (2001), 253–271 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Nao12]
    A. Naor, An introduction to the Ribe program, Japanese Journal of Mathematics 7 (2012), 167–233.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [NPSS06]
    A. Naor, Y. Peres, O. Schrammand S. Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Mathematical Journal 134 (2006), 165–197.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [NRS05]
    A. Naor, Y. Rabani and A. Sinclair, Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs, Journal of Functional Analysis 227 (2005), 273–303.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [Pis79]
    G. Pisier, Estimations des distances à un espace euclidien et des constantes de projection des espaces de Banach de dimension finie; d’après H. König et al., in Séminaire d’Analyse Fonctionnelle (1978–1979), École Polytech., Palaiseau, 1979, Exp. No. 10, 21.Google Scholar
  55. [Rut65]
    D. Rutovitz, Some parameters associated with finite-dimensional Banach spaces, Journal of the London Mathematical Society 40 (1965), 241–255.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [Vil03]
    C. Villani, Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003.CrossRefzbMATHGoogle Scholar
  57. [Woj91]
    P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, Vol. 25, Cambridge University Press, Cambridge, 1991.CrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.The Rachel and Selim Benin School of Computer Science and EngineeringThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations