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

@article{Azzam2016AnAT,
  title={An analyst’s traveling salesman theorem for sets of dimension larger than one},
  author={Jonas Azzam and Raanan Schul},
  journal={Mathematische Annalen},
  year={2016},
  volume={370},
  pages={1389-1476}
}
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 measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to… 
A $d$-dimensional Analyst's Travelling Salesman Theorem for subsets of Hilbert space
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
Harmonic Measure and the Analyst's Traveling Salesman Theorem
We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose
Sets with topology, the Analyst's TST, and applications
This paper was motivated by three questions. First: in a recent paper, Azzam and Schul asked what sort of sets could play the role of curves in the context of the higher dimensional analyst's
An Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$
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 +
Subsets of rectifiable curves in Banach spaces: sharp exponents in Schul-type theorems
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
Dahlberg's theorem in higher co-dimension
Quantitative comparisons of multiscale geometric properties
We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and
Geometry of Measures in Real Dimensions via Hölder Parameterizations
We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $$\mathbb {R}^n$$Rn when s is a real number between 0 and n. This topic
Identifying 1-rectifiable measures in Carnot groups
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem
...
1
2
3
4
...

References

SHOWING 1-10 OF 35 REFERENCES
Rectifiable sets and the Traveling Salesman Problem
Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image
Subsets of rectifiable curves in Hilbert space-the analyst’s TSP
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
Characterization of n-rectifiability in terms of Jones’ square function: Part II
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
Quantitative Reifenberg theorem for measures
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
Multiscale analysis of 1-rectifiable measures: necessary conditions
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
LEAST SQUARES APPROXIMATIONS OF MEASURES VIA GEOMETRIC CONDITION NUMBERS
For a probability measure μ on a real separable Hilbert space H , we are interested in “volume-based” approximations of the d -dimensional least squares error of μ , i.e., least squares error with
Multiscale Analysis of 1-rectifiable Measures II: Characterizations
Abstract 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
Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps
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
Two sufficient conditions for rectifiable measures
We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending
The Dimension of the Brownian Frontier Is Greater Than 1
Consider a planar Brownian motion run for finite time. Thefrontieror “outer boundary” of the path is the boundary of the unbounded component of the complement. K. Burdzy (Prob. Theory Related
...
1
2
3
4
...