Journal d'Analyse Mathématique

, Volume 103, Issue 1, pp 331–375 | Cite as

Subsets of rectifiable curves in Hilbert space-the analyst’s TSP



We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in ℝd. Their results formed the basis of quantitative rectifiability in ℝd. We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales aroundmost points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analogue of the Square function. This allows us to conclude that for a given set K, the ℓ2 norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90, Oki92] are independent of the ambient dimension.


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  1. [Aro03]
    S. Arora, Approximation schemes for NP-hard geometric optimization problems: a survey, Math. Program. 97 (2003), Ser. B, 43–69.MATHMathSciNetGoogle Scholar
  2. [BJ90]
    C. J. Bishop and P. W. Jones, Harmonic measure and arclength, Ann. of Math. (2) 132 (1990), 511–547.CrossRefMathSciNetGoogle Scholar
  3. [Chr90]
    M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601–628.MathSciNetGoogle Scholar
  4. [Dav91]
    G. David, Wavelets and Singular Integrals on Curves and Surfaces, Lecture Notes in Mathematics 1485, Springer-Verlag, Berlin, 1991.Google Scholar
  5. [DS93]
    G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Ame. Math. Soc., Providence, RI, 1993.MATHGoogle Scholar
  6. [FFPar]
    F. Ferrari, B. Franchi, and H. Pajot, The geometric traveling salesman theorem in the Heisenberg group, Rev. Mat. Iberoamericana 23 (2007), 437–480.MATHMathSciNetGoogle Scholar
  7. [Hah05]
    I. Hahlomaa, Menger curvature and Lipschitz parametrizations in metric spaces, Fund.Math. 185 (2005), 143–169.MATHMathSciNetGoogle Scholar
  8. [Hahar]
    I. Hahlomaa, Curvature integral and Lipschitz parametrizations in 1-regular metric spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), 99–123.MATHMathSciNetGoogle Scholar
  9. [JM02]
    D. S. Johnson and L. A. McGeoch, Experimental analysis of heuristics for the STSP, in The Traveling Salesman Problem and its Variations, Kluwer Acad. Publ., Dordrecht, 2002, pp. 369–443.Google Scholar
  10. [Jon88]
    P. W. Jones, Lipschitz and bi-Lipschitz functions, Rev. Mat. Iberoamericana 4 (1988), 115–121.MATHMathSciNetGoogle Scholar
  11. [Jon90]
    P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), 1–15.MATHCrossRefMathSciNetGoogle Scholar
  12. [KK92]
    C. Kenyon and R. Kenyon, How to take short cuts, Discrete Comput. Geom. 8 (1992), 251–264.MATHCrossRefMathSciNetGoogle Scholar
  13. [Ler03]
    G. Lerman, Quantifying curvelike structures of measures by using L 2 Jones quantities, Comm. Pure Appl. Math. 56 (2003), 1294–1365.MATHCrossRefMathSciNetGoogle Scholar
  14. [Mat95]
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge University Press, Cambridge, 1995.MATHGoogle Scholar
  15. [Oki92]
    K. Okikiolu, Characterization of subsets of rectifiable curves in Rn, J. London Math. Soc. (2) 46 (1992), 336–348.MATHCrossRefMathSciNetGoogle Scholar
  16. [Paj02]
    H. Pajot, Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral, Lecture Notes in Mathematics 1799, Springer-Verlag, Berlin, 2002.MATHGoogle Scholar
  17. [Sch05]
    R. Schul, Subsets of Rectifable Curves in Hilbert Space and the Analyst’s TSP, PhD thesis, Yale University, 2005.Google Scholar
  18. [Schara]
    R. Schul, Ahlfors-regular curves in metric spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), 431–460.MathSciNetGoogle Scholar
  19. [Scharb]
    R. Schul, Analyst’s traveling salesman theorems. A survey, in In the Tradition of Ahlfors-Bers IV, Contemp. Math. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 209–220.Google Scholar
  20. [SS83]
    E. M. Stein and J.-O. Stromberg, Behavior of maximal functions in R n for large n, Ark.Mat. 21 (1983), 259–269.MATHCrossRefMathSciNetGoogle Scholar
  21. [Ste83]
    E. M. Stein, Some results in harmonic analysis in R n, for n → ∞, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 71–73.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUCLALos Angeles

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