The Traveling Salesman Theorem in Carnot groups

Chousionis, Vasilis; Li, Sean; Zimmerman, Scott

doi:10.1007/s00526-018-1446-3

Cited by 15 publications

(36 citation statements)

References 27 publications

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“…Similar results have been considered in a number of non-Hilbert space settings, for example, the Heisenberg group [FFP + 07, LS16b, LS16a], Carnot groups [CLZ19,Li19], and general metric spaces [Hah05,Hah08,Sch07a,DS20]. See [BNV19] for a similar result concerning Hölder curves.…”

Section: Introductionsupporting

confidence: 63%

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

Hyde¹

2021

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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: Introductionsupporting

confidence: 63%

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

Hyde¹

2021

Preprint

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“…The original theorem in Jones's paper has γ expressed in terms of a sum over dyadic cubes of R 2 (and β is also expressed in terms of cubes), but it is well known that the sum over cubes is equivalent to the integral over balls up to absolute multiplicative constants. It is also important to note that the 2 in the exponent of β 2 comes from the power type of the modulus of convexity of R n whereas the 2 in the exponent of the r 2 is simply the Hausdorff dimension of R 2 .…”

Section: Theorem 11 ([7])mentioning

confidence: 99%

“…As for addressing the analyst's traveling salesman problem in non-Euclidean spaces, the majority of the effort has been in the setting of Carnot groups [2], and in particular the Heisenberg group [4,8,10,11] (although there has also been work done in certain fractal spaces [3]).…”

Section: Theorem 11 ([7])mentioning

confidence: 99%

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Stratified $β$-numbers and traveling salesman in Carnot groups

Li¹

2019

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We introduce a modified version of P. Jones's β-numbers for Carnot groups which we call stratified β-numbers. We show that an analogue of Jones's traveling salesman theorem on 1-rectifiability of sets holds for any Carnot group if we replace previous notions of β-numbers in Carnot groups with stratified β-numbers. In particular, we can generalize both directions of the traveling salesman theorem giving us a characterization of subsets of Carnot groups that lie on finite length rectifiable curves. Our proof expands upon previous analysis of the Hebisch-Sikora norm for Carnot groups. In particular, we find new estimates on the drift between almost parallel line segments that take advantage of the stratified β's and also develop a Taylor expansion technique of the norm.

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“…Underpinning the main theorem is a characterization of subsets of rectifiable curves, with estimates on the length of the shortest curve containing a given set, usually called the analyst's traveling salesman theorem. First established in R n by Jones [43], when n = 2, and by Okikiolu [59], when n ≥ 3, the analyst's traveling salesman theorem was recently extended to arbitrary Carnot groups by the second author [49] (for earlier necessary or sufficient conditions, see [20,33,44,50,51]). A key insight in [49] is that to obtain a full characterization of subsets of rectifiable curves, with effective estimates on length, the local deviation of the set from a horizontal line should incorporate distance in each layer of the Carnot group.…”

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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The Traveling Salesman Theorem in Carnot groups

Cited by 15 publications

References 27 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

Stratified $β$-numbers and traveling salesman in Carnot groups

Identifying 1-rectifiable measures in Carnot groups

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