A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$

Hyde, Matthew

doi:10.48550/arxiv.2006.16677

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…Azzam and Schul's proof of the above result relies heavily on the use of the Besicovitch Covering Theorem and is unsuitable in our setting. Our proof of Lemma 2.13 follows a similar approach to the proof of [Hyd20,Lemma 2.29]. An important ingredient is the following.…”

Section: Preliminariesmentioning

confidence: 97%

“…Stable d-surfaces were defined by David in [Dav04]. In [Hyd20], we introduce a new β-number and show that for any set E and Q 0 ∈ D, Q 0 can be contained in a lower regular set F with βE,C0,p (Q 0 ) ∼ β F,C0,p (Q 0 ), where βE,C0,p is defined as in the statement of Theorem 1.6 with respect to the β-number from [Hyd20]. Combing the two, for any E ⊆ R n and Q 0 ∈ D, there exists a stable d-surface Σ so that Q 0 ⊆ Σ and H d (Σ) ∼ βE,C0,p (Q 0 ).…”

Section: Introductionmentioning

confidence: 99%

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

Hyde¹

2021

Preprint

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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 Theorem in this setting by showing for any lower d-regularwhere β d (E) give a measure of the curvature of E and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes).To do this, we show how to modify the Reifenberg Parametrization Theorem of David and Toro so that it holds in Hilbert space. As a corollary, we show that a set E ⊆ H is uniformly rectifiable if and only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that E is bi-laterally well approximated by planes at most scales and locations. Contents 1. Introduction 1 2. Preliminaries 5 3. Reifenberg parametrisations in Hilbert space 12 4. Estimating curvature for Reifenberg flat sets 16 5. Estimating curvature for general sets 23 6. Estimating measure 32 7. Bilateral Weak Geometric Lemma 38 Appendix A. Parameterisation in Hilbert space 43 Appendix B. Constants 54 Appendix C. Reduction to Euclidean space 56 References 61

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Section: Preliminariesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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

Hyde¹

2021

Preprint

Self Cite

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“…Also see [46]. Progress on traveling salesman theorems for higher-dimensional objects has been made in [7,12,40,64].…”

Section: Introductionmentioning

confidence: 99%

Identifying 1-rectifiable measures in Carnot groups

Badger¹,

Li²,

Zimmerman³

2021

Preprint

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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 of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R 2 (P. Jones, 1990), in R n (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' β-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.

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A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$

Cited by 2 publications

References 16 publications

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

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

Identifying 1-rectifiable measures in Carnot groups

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