# Effective Reifenberg theorems in Hilbert and Banach spaces

@article{Edelen2018EffectiveRT, title={Effective Reifenberg theorems in Hilbert and Banach spaces}, author={Nick Edelen and Aaron Naber and Daniele Valtorta}, journal={Mathematische Annalen}, year={2018}, pages={1-80} }

A famous theorem by Reifenberg states that closed subsets of $$\mathbb {R}^n$$Rn that look sufficiently close to k-dimensional at all scales are actually $$C^{0,\gamma }$$C0,γ equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure $$\mu $$μ in $$\mathbb {R}^n$$Rn, one may introduce the k-dimensional Jones’ $$\beta $$β-numbers of the measure, where $$\beta ^k_\mu (x,r)$$βμk(x,r) quantifies on a given ball $$B_r(x)$$Br(x… Expand

#### 14 Citations

Sufficient conditions for $C^{1,\alpha}$ parametrization and rectifiability

- Mathematics
- 2017

We say a measure is $C^{1,\alpha}$ $d$-rectifiable if there is a countable union of $C^{1,\alpha}$ $d$-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure… Expand

Subsets of rectifiable curves in Banach spaces: sharp exponents in Schul-type theorems

- Mathematics
- 2020

The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and… Expand

An Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$

- Physics, Mathematics
- 2020

In his 1990 paper, Jones proved the following: given $E \subseteq \mathbb{R}^2$, there exists a curve $\Gamma$ such that $E \subseteq \Gamma$ and \[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E +… Expand

A $d$-dimensional Analyst's Travelling Salesman Theorem for subsets of Hilbert space

- Mathematics
- 2021

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. We prove a version of Azzam and Schul’s d-dimensional Analyst’s Travelling Salesman… Expand

Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

- Mathematics
- 2019

In this article we extend to arbitrary p -energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $$p=2$$ p = 2 . We first show that the set of… Expand

A sharp necessary condition for rectifiable curves in metric spaces

- Mathematics
- 2019

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a… Expand

A remark on two notions of flatness for sets in the Euclidean space

- Mathematics
- 2021

In this note we compare two ways of measuring the n-dimensional “flatness” of a set S ⊂ R, where n ∈ N and d > n. The first one is to consider the classical Reifenberg-flat numbers α(x, r) (x ∈ S, r… Expand

Quantitative estimates for the singular strata of minimizing Harmonic maps

- Mathematics
- 2018

Regularity properties for minimizing harmonic maps between Riemannian
manifolds have been known since the classical work of Schoen and Uhlenbeck (1982); in that context, an estimate on the Hausdorff… Expand

Estimates on the generalized critical strata of Green's function

- Mathematics
- 2019

In this paper, we obtain quantitative estimates on the fine structure of Green's functions for pairs of complementary domains, $\Omega^+, \Omega^- \subset \mathbb{R}^n$ which arise in a class of… Expand

Sufficient conditions for C^1,α parametrization and rectifiability

- Mathematics
- Annales Academiae Scientiarum Fennicae Mathematica
- 2020

We say a measure is C d-rectifiable if there is a countable union of C d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in R to be C d-rectifiable,… Expand

#### References

SHOWING 1-10 OF 51 REFERENCES

Multiscale analysis of 1-rectifiable measures: necessary conditions

- Mathematics
- 2015

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $$\mathbb {R}^n$$Rn, $$n\ge 2$$n≥2. To each locally finite Borel measure $$\mu… Expand

Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps

- Mathematics
- 2015

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is… Expand

Characterization of n-rectifiability in terms of Jones’ square function: Part II

- Mathematics
- 2015

We show that a Radon measure $${\mu}$$μ in $${\mathbb{R}^d}$$Rd which is absolutely continuous with respect to the n-dimensional Hausdorff measure $${\mathcal{H}^n}$$Hn is n-rectifiable if the so… Expand

Quantitative Reifenberg theorem for measures

- Mathematics
- 2016

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $\beta$-numbers, which appropriately measure how close the support is to being contained… Expand

Characterization of n-rectifiability in terms of Jones’ square function: part I

- Mathematics
- 2015

In this paper it is shown that if $$\mu $$μ is a finite Radon measure in $${\mathbb R}^d$$Rd which is n-rectifiable and $$1\le p\le 2$$1≤p≤2, then $$\begin{aligned} \displaystyle \int _0^\infty \beta… Expand

Reifenberg Parameterizations for Sets with Holes

- Mathematics
- 2009

We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of… Expand

An analyst’s traveling salesman theorem for sets of dimension larger than one

- Mathematics
- 2016

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $$\beta $$β-numbers. These $$\beta $$β-numbers are geometric quantities… Expand

An upper bound for the length of a Traveling Salesman path in the Heisenberg group

- Mathematics
- 2014

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) to be contained in a rectifiable curve is that it satisfies a modified… Expand

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

- Mathematics
- 2006

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… Expand

Solution of the Plateau problem form-dimensional surfaces of varying topological type

- Mathematics
- 1960

We use a definition due to J. F. Adams: DEFINITION. Let G be a compact Abelian group. Let S be a closed set in N-dimensional Euclidean space and A a closed subset of S. Let m be a non-negative… Expand