# Effective Reifenberg theorems in Hilbert and Banach spaces

## Abstract

A famous theorem by Reifenberg states that closed subsets of $$\mathbb {R}^n$$ that look sufficiently close to k-dimensional at all scales are actually $$C^{0,\gamma }$$ equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure $$\mu$$ in $$\mathbb {R}^n$$, one may introduce the k-dimensional Jones’ $$\beta$$-numbers of the measure, where $$\beta ^k_\mu (x,r)$$ quantifies on a given ball $$B_r(x)$$ how closely in an integral sense the support of the measure is to living inside a k-dimensional subspace. Recently, it has been proven that if these $$\beta$$-numbers satisfy the uniform summability estimate $$\int _0^2 \beta ^k_\mu (x,r)^2 \frac{dr}{r}<M$$, then $$\mu$$ must be rectifiable with uniform measure bounds. Note that one only needs the square of the $$\beta$$-numbers to satisfy the summability estimate, this power gain has played an important role in the applications, for instance in the study of singular sets of geometric equations. One may also weaken these pointwise summability bounds to bounds which are more integral in nature. The aim of this article is to study these effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from $$\mathbb {R}^n$$ continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a weak version of the effective Reifenberg theorem holds in that if one assumes a summability estimate $$\int _0^2 \beta ^k_\mu (x,r)^1 \frac{dr}{r}<M$$without power gain, then $$\mu$$ must again be rectifiable with measure estimates. Improving this estimate in order to obtain a power gain turns out to be a subtle issue. For $$k=1$$ we will see for a uniformly smooth Banach space that if $$\int _0^2 \beta ^1_\mu (x,r)^\alpha \frac{dr}{r}<M^{\alpha /2}$$, where $$\alpha$$ is the smoothness power of the Banach space, then $$\mu$$ is again rectifiable with uniform measure estimates.

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We mention that since X is not assumed to be separable, the complement of the support of $$\mu$$ may not have measure zero, so it’s better to talk about sets of full measure rather than supports.

## References

1. 1.

Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math., vol. 178, pp. 15–50. Dekker, New York (1996)

2. 2.

Alber, Y.I.: Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces. Appl. Math. Lett. 11, 115–121 (1998). https://doi.org/10.1016/S0893-9659(98)00112-8

3. 3.

Azzam, J., Schul, R.: An analyst’s traveling salesman theorem for sets of dimension larger than one. Math. Ann. 370, 1389–1476 (2018). https://doi.org/10.1007/s00208-017-1609-0. (Available at arXiv:1609.02892)

4. 4.

Azzam, J., Tolsa, X.: Characterization of $$n$$-rectifiability in terms of Jones’ square function: Part II. Geom. Funct. Anal. 25, 1371–1412 (2015). https://doi.org/10.1007/s00039-015-0334-7

5. 5.

Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann. 361, 1055–1072 (2015). https://doi.org/10.1007/s00208-014-1104-9

6. 6.

Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures II: characterizations. Anal. Geom. Metr. Spaces 5, 1–39 (2017). https://doi.org/10.1515/agms-2017-0001. Available at arXiv:1602.03823

7. 7.

Bishop, C.J., Peres, Y.: Fractals in Probability and Analysis. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/9781316460238. (English)

8. 8.

Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc. 40, 396–414 (1936). https://doi.org/10.2307/1989630

9. 9.

David, G., Semmes, S.: Singular integrals and rectifiable sets in $$\mathbb{R}^n$$: beyond Lipschitz graphs, Astérisque, vol. 191. Sociètè Mathèmatique de France, Paris (1991)

10. 10.

David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence (1993). https://doi.org/10.1090/surv/038

11. 11.

David, G., Toro, T.: Reifenberg parameterizations for sets with holes. Mem. Am. Math. Soc. 215, vi+102 (2012). https://doi.org/10.1090/S0065-9266-2011-00629-5

12. 12.

David, G.C., Schul, R.: The analyst’s traveling salesman theorem in graph inverse limits. Ann. Acad. Sci. Fenn. Math. 42, 649–692 (2017). https://doi.org/10.5186/aasfm.2017.4260

13. 13.

Edelen, N.S., Naber, A., Valtorta, D.: Quantitative Reifenberg theorem for measures. arXiv:1612.08052

14. 14.

Federer, H.: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)

15. 15.

Ferrari, F., Franchi, B., Pajot, H.: The geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam. 23, 437–480 (2007). https://doi.org/10.4171/RMI/502

16. 16.

Garnett, J., Killip, R., Schul, R.: A doubling measure on $$\mathbb{R}^d$$ can charge a rectifiable curve. Proc. Am. Math. Soc. 138, 1673–1679 (2010)

17. 17.

Hahlomaa, I.: Menger curvature and rectifiability in metric spaces. Adv. Math. 219, 1894–1915 (2008). https://doi.org/10.1016/j.aim.2008.07.013

18. 18.

Hanner, O.: On the uniform convexity of $$L^p$$ and $$l^p$$. Ark. Mat. 3, 239–244 (1956). https://doi.org/10.1007/BF02589410

19. 19.

Hudzik, H., Wang, Y., Sha, R.: Orthogonally complemented subspaces in Banach spaces. Numer. Funct. Anal. Optim. 29, 779–790 (2008). https://doi.org/10.1080/01630560802279231

20. 20.

Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102, 1–15 (1990). https://doi.org/10.1007/BF01233418

21. 21.

Li, S., Schul, R.: The traveling salesman problem in the Heisenberg group: upper bounding curvature. Trans. Am. Math. Soc. 368, 4585–4620 (2016). https://doi.org/10.1090/tran/6501

22. 22.

Li, S., Schul, R.: An upper bound for the length of a traveling salesman path in the Heisenberg group. Rev. Mat. Iberoam. 32, 391–417 (2016). https://doi.org/10.4171/RMI/889

23. 23.

Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Springer, Berlin (1977). (Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92)

24. 24.

Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin (1979). (Function spaces)

25. 25.

Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995). https://doi.org/10.1017/CBO9780511623813. (Fractals and rectifiability)

26. 26.

Miśkiewicz, M.: Discrete Reifenberg-type theorem. Annales Academiae Scientiarum Fennicae. Mathematica 43, 3–19 (2018). https://doi.org/10.5186/aasfm.2018.4301

27. 27.

Naber, A., Valtorta, D.: Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Math. (2) 185, 131–227 (2017). https://doi.org/10.4007/annals.2017.185.1.3

28. 28.

Nordlander, G.: The modulus of convexity in normed linear spaces. Ark. Mat. 4(1960), 15–17 (1960). https://doi.org/10.1007/BF02591317

29. 29.

Okikiolu, K.: Characterization of subsets of rectifiable curves in $${ R }^n$$. J. Lond. Math. Soc. (2) 46, 336–348 (1992). https://doi.org/10.1112/jlms/s2-46.2.336

30. 30.

Parthasarathy, K.R.: Probability Measures on Metric Spaces. AMS Chelsea Publishing, Providence (2005). https://doi.org/10.1090/chel/352. (Reprint of the 1967 original)

31. 31.

Randrianantoanina, B.: Norm-one projections in Banach spaces. Taiwan. J. Math. 5, 35–95 (2001). https://doi.org/10.11650/twjm/1500574888. (International Conference on Mathematical Analysis and its Applications (Kaohsiung, 2000))

32. 32.

Reed, M., Simon, B.: I: Functional Analysis, Methods of Modern Mathematical Physics. Elsevier Science (1981). Available at https://books.google.com/books?id=rpFTTjxOYpsC

33. 33.

Reifenberg, E.R.: Solution of the plateau problem for $$m$$-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960). Available at http://link.springer.com/article/10.1007

34. 34.

Schul, R.: Analyst’s traveling salesman theorems. A survey. In: In the Tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, pp. 209–220. Amer. Math. Soc., Providence (2007). https://doi.org/10.1090/conm/432/08310

35. 35.

Schul, R.: Subsets of rectifiable curves in Hilbert space—the analyst’s TSP. J. Anal. Math. 103, 331–375 (2007). https://doi.org/10.1007/s11854-008-0011-y

36. 36.

Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983)

37. 37.

Tolsa, X.: Rectifiability of measures and the $$\beta _p$$ coefficients. arXiv:1708.02304 (preprint)

38. 38.

Tolsa, X.: Characterization of $$n$$-rectifiability in terms of Jones’ square function: part I. Calc. Var. Partial Diff. Equ. 54, 3643–3665 (2015). https://doi.org/10.1007/s00526-015-0917-z

39. 39.

Toro, T.: Geometric conditions and existence of bi-Lipschitz parameterizations. Duke Math. J. 77, 193–227 (1995). https://doi.org/10.1215/S0012-7094-95-07708-4

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Correspondence to Nick Edelen.