The analyst's traveling salesman theorem in graph inverse limits

David, Guy C.; Schul, Raanan

doi:10.5186/aasfm.2017.4260

Cited by 14 publications

(6 citation statements)

References 25 publications

(93 reference statements)

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“…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%

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

Li¹

2019

Preprint

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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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Section: Theorem 11 ([7])mentioning

confidence: 99%

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

Li¹

2019

Preprint

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“…For more information, we refer the reader to the books [9,25,31] There have been numerous generalizations and variants of Theorem 1.1 beyond Euclidean spaces. Schul [27] extended Theorem 1.1 to Hilbert spaces, David and Schul [11] recently considered the theorem in the graph inverse limits of Cheeger-Kleiner, and Hahlomaa and Schul (independently) [15,16,26] obtained variants of Theorem 1.1 in general metric spaces. In the last case, however, there is no natural notion of lines over which one may infimize in the definition of β, so curvature-type quantities other than β-numbers must be considered.…”

Section: Introductionmentioning

confidence: 99%

The Traveling Salesman Theorem in Carnot groups

Chousionis

Zimmerman

2018

Calc. Var.

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Let G be any Carnot group. We prove that, if a subset of G is contained in a rectifiable curve, then it satisfies Peter Jones' geometric lemma with some natural modifications. We thus prove one direction of the Traveling Salesman Theorem in G. Our proof depends on new Alexandrov-type curvature inequalities for the Hebisch-Sikora metrics. We also apply the geometric lemma to prove that, in every Carnot group, there exist −1-homogeneous Calderón-Zygmund kernels such that, if a set E ⊂ G is contained in a 1-regular curve, then the corresponding singular integral operators are bounded in L 2 (E). In contrast to the Euclidean setting, these kernels are nonnegative and symmetric.

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“…Schul managed to give an analogue of Okikiolu's and Jones's results in Hilbert Space by defining an infinite series of beta numbers reliant on "multiresolution families" rather than dyadic cubes; whereas the dyadic cubes are defined using the geometry of the ambient space R n , multiresolution families are collections of balls intrinsic to a given set [Sch07]. Further traveling salesman results have been proven for Radon measures [BS17] and in many different spaces such as Carnot groups [Li19], graph inverse limit spaces [DS16], Banach spaces [BM20], and general metric spaces [Hah05], [DS20].…”

Section: Introduction 1overviewmentioning

confidence: 97%

The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

Krandel¹

2021

Preprint

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Let Γ ⊆ H be a Jordan arc in a Hilbert space H with distance between its endpoints given by crd(Γ) and length given by (Γ). We provewhere Ĝ is a multiresolution family of balls centered on Γ. This provides a sharpening of Schul's original Hilbert space traveling salesman theorem analogous to Bishop's sharpening of the traveling salesman theorem in R n in the case of a Jordan arc. Specifically, the most significant improvement comes from strengthening Schul's result in the case of a Jordan arc by replacing (Γ) in the direction with (Γ) − crd(Γ).

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The analyst's traveling salesman theorem in graph inverse limits

Cited by 14 publications

References 25 publications

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

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

The Traveling Salesman Theorem in Carnot groups

The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

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