# 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}
}
• Published 9 September 2016
• Mathematics
• Mathematische Annalen
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…
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## 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
• 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
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
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
LEAST SQUARES APPROXIMATIONS OF MEASURES VIA GEOMETRIC CONDITION NUMBERS
• Mathematics
• 2012
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
• Mathematics
• 2016
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
• 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$is Two sufficient conditions for rectifiable measures • Mathematics • 2014 We identify two sufficient conditions for locally finite Borel measures on$\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
• Mathematics
• 1995
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