Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

Open Access
See all formats and pricing
More options …

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger
  • Corresponding author
  • Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, United States of America
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Raanan Schul
Published Online: 2017-03-16 | DOI: https://doi.org/10.1515/agms-2017-0001


A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.

Keywords: 1 rectifiable measures; purely 1 unrectifiable measures; rectifiable curves; Jones beta numbers; Jones square functions; Analyst’s traveling salesman theorem; doubling measures; Hausdorff densities; Hausdorff measures


  • [1] L. Ambrosio and B. Kirchheim. Rectifiable sets in metric and Banach spaces. Math. Ann., 318(3):527-555, 2000.Google Scholar

  • [2] J. Azzam, G. David, and T. Toro. Wasserstein distance and the rectifiability of doubling measures: part I. Math. Ann., 364(1- 2):151-224, 2016.Google Scholar

  • [3] J. Azzam and M. Mourgoglou. A characterization of 1-rectifiable doubling measures with connected supports. Anal. PDE, 9(1):99-109, 2016.Google Scholar

  • [4] J. Azzam and X. Tolsa. Characterization of n-rectifiability in terms of Jones’ square function: Part II. Geom. Funct. Anal., 25(5):1371-1412, 2015.CrossrefGoogle Scholar

  • [5] M. Badger and R. Schul. Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann., 361(3-4):1055-1072, 2015.Google Scholar

  • [6] M. Badger and R. Schul. Two suficient conditions for rectifiable measures. Proc. Amer.Math. Soc., 144(6):2445-2454, 2016.Google Scholar

  • [7] D. Bate. Structure of measures in Lipschitz difierentiability spaces. J. Amer. Math. Soc., 28(2):421-482, 2015.Google Scholar

  • [8] D. Bate and S. Li. Characterizations of rectifiable metric measure spaces. preprint, arXiv:1409.4242, to appear in Ann. Sci. Éc. Norm. Supèr., 2014.Google Scholar

  • [9] G. Beer. Topologies on closed and closed convex sets, volume 268 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.Google Scholar

  • [10] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann., 98(1):422-464, 1928.Google Scholar

  • [11] A. S. Besicovitch. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann., 115(1):296-329, 1938.Google Scholar

  • [12] V. Chousionis and J. T. Tyson. Marstrand’s density theorem in the Heisenberg group. Bull. Lond. Math. Soc., 47(5):771-788, 2015.Google Scholar

  • [13] G. David and S. Semmes. Singular integrals and rectifiable sets in Rn: Beyond Lipschitz graphs. Astérisque, (193):152, 1991.Google Scholar

  • [14] G. David and S. Semmes. Analysis of and on uniformly rectifiable sets, volume 38 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1993.Google Scholar

  • [15] G. David and T. Toro. Reifenberg parameterizations for sets with holes. Mem. Amer. Math. Soc., 215(1012):vi+102, 2012.Google Scholar

  • [16] G. C. David and R. Schul. The Analyst’s traveling salesman theorem in graph inverse limits. preprint, 2016.Google Scholar

  • [17] C. De Lellis. Recti_able sets, densities and tangent measures. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.Google Scholar

  • [18] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015.Google Scholar

  • [19] K. J. Falconer. The geometry of fractal sets, volume 85 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1986.Google Scholar

  • [20] H. Federer. The (', k) rectifiable subsets of n-space. Trans. Amer. Soc., 62:114-192, 1947.Google Scholar

  • [21] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.Google Scholar

  • [22] F. Ferrari, B. Franchi, and H. Pajot. The geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 23(2):437-480, 2007.CrossrefGoogle Scholar

  • [23] J. Garnett, R. Killip, and R. Schul. A doubling measure on Rd can charge a rectifiable curve. Proc. Amer. Math. Soc., 138(5):1673-1679, 2010.Google Scholar

  • [24] I. Hahlomaa. Menger curvature and rectifiability in metric spaces. Adv. Math., 219(6):1894-1915, 2008.Google Scholar

  • [25] P. Hajłasz and S. Malekzadeh. On conditions for unrectifiability of a metric space. Anal. Geom. Metr. Spaces, 3:1-14, 2015.Google Scholar

  • [26] P.W. Jones. Square functions, Cauchy integrals, analytic capacity, and harmonic measure. In Harmonic analysis and partial differential equations (El Escorial, 1987), volume 1384 of Lecture Notes in Math., pages 24-68. Springer, Berlin, 1989.Google Scholar

  • [27] P. W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1-15, 1990.CrossrefGoogle Scholar

  • [28] P. W. Jones, G. Lerman, and R. Schul. The Analyst’s traveling bandit problem in Hilbert space. in preparation.Google Scholar

  • [29] N. Juillet. A counterexample for the geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 26(3):1035-1056, 2010.CrossrefGoogle Scholar

  • [30] B. Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc., 121(1):113-123, 1994.Google Scholar

  • [31] O. Kowalski and D. Preiss. Besicovitch-type properties of measures and submanifolds. J. Reine Angew. Math., 379:115-151, 1987.Google Scholar

  • [32] J. C. Léger. Menger curvature and rectifiability. Ann. of Math. (2), 149(3):831-869, 1999.Google Scholar

  • [33] G. Lerman. Quantifying curvelike structures of measures by using L2 Jones quantities. Comm. Pure Appl.Math., 56(9):1294- 1365, 2003.Google Scholar

  • [34] G. Lerman and J. T. Whitehouse. High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities. Rev. Mat. Iberoam., 27(2):493-555, 2011.CrossrefGoogle Scholar

  • [35] S. Li and R. Schul. An upper bound for the length of a traveling salesman path in the Heisenberg group. Rev. Mat. Iberoam. 32(2):391-417, 2016.CrossrefGoogle Scholar

  • [36] S. Li and R. Schul. The traveling salesman problem in the Heisenberg group: Upper bounding curvature. Trans. Amer.Math. Soc., 368(7):4585-4620, 2016.Google Scholar

  • [37] A. Lorent. Rectifiability of measures with locally uniform cube density. Proc. London Math. Soc. (3), 86(1):153-249, 2003.Google Scholar

  • [38] A. Lorent. A Marstrand type theorem for measures with cube density in general dimension. Math. Proc. Cambridge Philos. Soc., 137(3):657-696, 2004.Google Scholar

  • [39] J. M. Marstrand. Hausdorff two-dimensional measure in 3-space. Proc. London Math. Soc. (3), 11:91-108, 1961.Google Scholar

  • [40] J. M. Marstrand. The (', s) regular subsets of n-space. Trans. Amer. Math. Soc., 113:369-392, 1964.Google Scholar

  • [41] H.Martikainen and T. Orponen. Boundedness of the density normalised Jones’ square function does not imply 1-rectifiability. preprint, arXiv:1605.04091, 2016.Google Scholar

  • [42] P. Mattila. Hausdorff m regular and rectifiable sets in n-space. Trans. Amer. Math. Soc., 205:263-274, 1975.Google Scholar

  • [43] P. Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability.Google Scholar

  • [44] E. F. Moore. Density ratios and (Φ, 1) recti_ability in n-space. Trans. Amer. Math. Soc., 69:324-334, 1950.Google Scholar

  • [45] A. P. Morse and J. F. Randolph. The Φ rectifiable subsets of the plane. Trans. Amer. Math. Soc., 55:236-305, 1944.Google Scholar

  • [46] A. Naber and D. Valtorta. Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Of Math. (2), 185(1):131-227, 2017.Google Scholar

  • [47] A. D. Nimer. A sharp bound on the Hausdorff dimension of the singular set of an n-uniform measure. preprint, arXiv:1510.03732, 2015.Google Scholar

  • [48] A. D. Nimer. Conical 3-uniform measures: characterizations & new examples. preprint, arXiv:1608.02604, 2016.Google Scholar

  • [49] K. Okikiolu. Characterization of subsets of rectifiable curves in Rn. J. London Math. Soc. (2), 46(2):336-348, 1992.Google Scholar

  • [50] H. Pajot. Conditions quantitatives de rectifiabilité. Bull. Soc. Math. France, 125(1):15-53, 1997.Google Scholar

  • [51] H. Pajot. Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, volume 1799 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.Google Scholar

  • [52] D. Preiss. Geometry of measures in Rn: distribution, rectifiability, and densities. Ann. of Math. (2), 125(3):537-643, 1987.Google Scholar

  • [53] D. Preiss and J. Tišer. On Besicovitch’s 1 2 -problem. J. London Math. Soc. (2), 45(2):279-287, 1992.Google Scholar

  • [54] C. A. Rogers. Hausdorff measures. CambridgeMathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1970 original, With a foreword by K. J. Falconer.Google Scholar

  • [55] R. Schul. Subsets of rectifiable curves in Hilbert space-the analyst’s TSP. J. Anal. Math., 103:331-375, 2007.Google Scholar

  • [56] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.Google Scholar

  • [57] X. Tolsa. Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, volume307 of Progress in Mathematics. Birkhäuser/Springer, Cham, 2014.Google Scholar

  • [58] X. Tolsa. Characterization of n-rectifiability in terms of Jones’ square function: part I. Calc. Var. Partial Differential Equations, 54(4):3643-3665, 2015.Google Scholar

  • [59] X. Tolsa. Uniform measures and uniform rectifiability. J. Lond. Math. Soc. (2), 92(1):1-18, 2015. Google Scholar

About the article

Received: 2016-07-25

Revised: 2017-01-22

Published Online: 2017-03-16

Citation Information: Analysis and Geometry in Metric Spaces, Volume 5, Issue 1, Pages 1–39, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2017-0001.

Export Citation

© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Nick Edelen, Aaron Naber, and Daniele Valtorta
Mathematische Annalen, 2018
Matthew Badger and Vyron Vellis
The Journal of Geometric Analysis, 2018
Henri Martikainen and Tuomas Orponen
Journal de Mathématiques Pures et Appliquées, 2017

Comments (0)

Please log in or register to comment.
Log in