Skip to main content
Log in

High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures

  • Published:
Constructive Approximation Aims and scope


We estimate d-dimensional least squares approximations of an arbitrary d-regular measure μ via discrete curvatures of d+2 variables. The main result bounds the least squares error of approximating μ (or its restrictions to balls) with a d-plane by an average of the discrete Menger-type curvature over a restricted set of simplices. Its proof is constructive and even suggests an algorithm for an approximate least squares d-plane. A consequent result bounds a multiscale error term (used for quantifying the approximation of μ with a sufficiently regular surface) by an integral of the discrete Menger-type curvature over all simplices. The preceding paper (part I) provided the opposite inequalities of these two results. This paper also demonstrates the use of a few other discrete curvatures which are different from the Menger-type curvature. Furthermore, it shows that a curvature suggested by Léger (Ann. Math. 149(3), pp. 831–869, 1999) does not fit within our framework.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Chen, G., Lerman, G.: Foundations of a multi-way spectral clustering framework for hybrid linear modeling. Found. Comput. Math. 9(5), 517–558 (2009)

    Article  MATH  Google Scholar 

  2. Chen, G., Lerman, G.: Spectral curvature clustering (SCC). Int. J. Comput. Vis. 81(3), 317–330 (2009)

    Article  Google Scholar 

  3. David, G.: Wavelets and Singular Integrals on Curves and Surfaces. Lecture Notes in Mathematics, vol. 1465. Springer, Berlin (1991)

    Google Scholar 

  4. David, G., Semmes, S.: Singular integrals and rectifiable sets in ℝn: au-delà des graphes Lipschitziens. Astérisque 193, 1–145 (1991)

    Google Scholar 

  5. David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets. Mathematics Surveys and Monographs, vol. 38. American Mathematical Society, Providence (1993)

    MATH  Google Scholar 

  6. Farag, H.M.: The Riesz kernels do not give rise to higher-dimensional analogues of the Menger–Melnikov curvature. Publ. Mat. 43(1), 251–260 (1999)

    MATH  MathSciNet  Google Scholar 

  7. Guillet, J.M., Lerman, G.: Numerical search for high-dimensional Menger–Riesz-type curvatures (in preparation)

  8. Hahlomaa, I.: Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math. 185(2), 143–169 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hahlomaa, I.: Curvature integral and Lipschitz parametrization in 1-regular metric spaces. Ann. Acad. Sci. Fenn. Math. 32(1), 99–123 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102(1), 1–15 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Léger, J.C.: Menger curvature and rectifiability. Ann. Math. (2) 149(3), 831–869 (1999)

    Article  MATH  Google Scholar 

  12. Lerman, G.: Quantifying curvelike structures of measures by using L 2 Jones quantities. Commun. Pure Appl. Math. 56(9), 1294–1365 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lerman, G., Whitehouse, J.T.: High-dimensional Menger-type curvatures—part I: Geometric multipoles and multiscale inequalities (submitted). See also

  14. Lerman, G., Whitehouse, J.T.: Least square approximations for probability distributions via multi-way curvatures (in preparation)

  15. Lerman, G., Whitehouse, J.T.: On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions. J. Approx. Theory 156, 52–81 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lerman, G., McQuown, J., Blais, A., Dynlacht, B.D., Chen, G., Mishra, B.: Functional genomics via multiscale analysis: application to gene expression and chip-on-chip data. Bioinformatics 23(3), 314–320 (2006)

    Article  Google Scholar 

  17. Lerman, G., McQuown, J., Mishra, B.: Multiscale robust regression and multiscale influence analysis (2009, submitted)

  18. Mattila, P., Melnikov, M., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. Math. (2) 144(1), 127–136 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Menger, K.: Untersuchungen über allgemeine Metrik. Math. Ann. 103(1), 466–501 (1930)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schul, R.: Ahlfors-regular curves in metric spaces. Ann. Acad. Sci. Fenn. Math. 32, 437–460 (2007)

    MATH  MathSciNet  Google Scholar 

  21. Schul, R.: Bi-Lipschitz decomposition of Lipschitz functions into a metric space. Rev. Mat. Iberoam. 25(2), 521–531 (2009)

    MATH  Google Scholar 

  22. Schul, R.: Big-pieces-of-Lipschitz-images implies a sufficient Carleson estimate in a metric space. arXiv:0706.2517 (2007)

  23. Semmes, S.: Bilipschitz embeddings of metric spaces into Euclidean spaces. Publ. Mat. 43(2), 571–653 (1999)

    MATH  MathSciNet  Google Scholar 

  24. Tolsa, X.: Principal values for Riesz transforms and rectifiability. J. Funct. Anal. 254(7), 1811–1863 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Väisälä, J.: Gromov hyperbolic spaces. Expo. Math. 23(3), 187–231 (2005)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gilad Lerman.

Additional information

Communicated by Mauro Maggioni.

This work has been supported by NSF grant #0612608

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lerman, G., Whitehouse, J.T. High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures. Constr Approx 30, 325–360 (2009).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification (2000)