# 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…

## 36 Citations

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

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

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- 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…

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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…

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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…

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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 +…

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- 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…

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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

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- 2017

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…

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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…

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