High-dimensional Menger-type curvatures. I: Geometric multipoles and multiscale inequalities. (English) Zbl 1232.28007

The authors define discrete and continuous Menger-type curvatures. They establish an upper bound on the continuous Menger-type curvature of an Ahlfors regular measure \(\mu\). As a consequence, they prove that a uniformly rectifiable measure satisfy a Carleson-type estimate in terms of the Menger-type curvature.


28A75 Length, area, volume, other geometric measure theory
42C99 Nontrigonometric harmonic analysis
60D05 Geometric probability and stochastic geometry
Full Text: DOI arXiv Euclid


[1] David, G.: Wavelets and singular integrals on curves and surfaces . Lecture Notes in Mathematics 1465 . Springer-Verlag, Berlin, 1991. · Zbl 0764.42019
[2] David, G. and Semmes, S.: Singular integrals and rectifiable sets in \(\mathbbR^n\): au-delà des graphes Lipschitziens. Astérisque 193 (1991), 152 pp. · Zbl 0743.49018
[3] David, G. and Semmes, S.: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs 38 . American Mathematical Society, Providence, RI, 1993. · Zbl 0832.42008
[4] Greengard, L. and Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73 (1987), no. 2, 325-348. · Zbl 0629.65005
[5] Gritzman, P. and Lassak, M.: Estimates for the minimal width of polytopes inscribed in convex bodies. Discrete Comput. Geom. 4 (1989), no. 1, 627-635. · Zbl 0692.52002
[6] Jones, P.W.: Characterizing one-dimensional uniform rectifiability of measures by their Menger curvature. Unpublished result, recorded by [Pajot, H.: Analytic capacity, rectifiability, Menger curvature and the Cauchy integral . Lecture Notes in Mathematics 1799 . Springer-Verlag, Berlin, 2002.]
[7] Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102 (1990), no. 1, 1-15. · Zbl 0731.30018
[8] Lerman, G. and Whitehouse, J.T.: High-dimensional Menger-type curvatures. Part II: \(d\)-separation and a menagerie of curvatures. Constr. Approx. 30 (2009), no. 3, 325-360. · Zbl 1222.28007
[9] Lerman, G. and Whitehouse, J.T.: Least squares approximations of measures via geometric condition numbers. Submitted to Mathematika . Current version is available at arxiv.org: · Zbl 1236.28001
[10] Lerman, G. and Whitehouse, J.T.: On \(d\)-dimensional \(d\)-semimetrics and simplex-type inequalities for high-dimensional sine functions. J. Approx. Theory 156 (2009), no. 1, 52-81. · Zbl 1170.46025
[11] Mattila, P., Melnikov, M. and Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. (2) 144 (1996), no. 1, 127-136. JSTOR: · Zbl 0897.42007
[12] Mattila, P. and Preiss, D.: Rectifiable measures in \(\mathbbR^n\) and existence of principal values for singular integrals. J. London Math. Soc. (2) 52 (1995), no. 3, 482-496. · Zbl 0880.28002
[13] Menger, K.: Untersuchungen über allgemeine Metrik. Math. Ann. 103 (1930), no. 1, 466-501. · JFM 56.0508.04
[14] Pajot, H.: Analytic capacity, rectifiability, Menger curvature and the Cauchy integral . Lecture Notes in Mathematics 1799 . Springer-Verlag, Berlin, 2002. · Zbl 1043.28002
[15] Schul, R.: Subsets of rectifiable curves in Hilbert space -the analyst’s TSP. J. Anal. Math. 103 (2007), 331-375. · Zbl 1152.28006
[16] Tolsa, X.: Uniform rectifiability, Calderón-Zygmund operators with odd kernel and quasiorthogonality. Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 393-426. · Zbl 1194.28005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.