The traveling salesman problem in the Heisenberg group: Upper bounding curvature

Li, Sean; Schul, Raanan

doi:10.1090/tran/6501

Cited by 31 publications

(91 citation statements)

References 17 publications

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“…Let γ : [0, 1] → Γ ⊂ X be a parametrization proportional to arc length. (See, e.g., Lemma 2.14 of [22] for the existence of this parametrization.) In particular, γ is Lipschitz with constant at most 32H 1 (Γ).…”

Section: The Curve the Balls And A Reductionmentioning

confidence: 99%

“…The idea of dividing the collection of balls into two sub-collections with these qualitative features is now standard (see [24], [27], [25], [22]) but our particular division is sensitive to the current context.…”

Section: The Filtrationsmentioning

confidence: 99%

“…To prove Proposition 6.1, we follow roughly the outline in Section 4 of [22]. This involves first proving some geometric results about the structure of arcs in G 2 -balls, and then using a geometric martingale argument, of the kind appearing in Section 4.2 of [22], as well as in [27], [25].…”

Section: Flat Ballsmentioning

confidence: 99%

“…We will follow the outline of Section 2.4 of [22], citing results from that paper when necessary. We observe that our results apply to m-adic scales whereas those results are stated in terms of dyadic scales, but this means only that the implied constants below depend on m.…”

Section: Flat Ballsmentioning

confidence: 99%

“…However, an example that shows that the analogue of part (A) is false appeared in [18]. In [22] it was shown that the analogue of part (A) does hold if one increases the exponent of β H from 2 to 4, and in [23] it was shown that part (B) holds for any exponent p < 4. (Note that by the definition of the Jones numbers, β H ≤ 1, so increasing the power reduces the quantity on the right-hand side of part (B).…”

Section: Introductionmentioning

confidence: 99%

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The analyst's traveling salesman theorem in graph inverse limits

David¹,

Schul²

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. We prove a version of Peter Jones' analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger-Kleiner. These spaces are constructed as inverse limits of metric graphs, and include examples which are doubling and have a Poincaré inequality. We show that a set in one of these spaces is contained in a rectifiable curve if and only if it is quantitatively "flat" at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces.

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Section: The Curve the Balls And A Reductionmentioning

confidence: 99%

Section: The Filtrationsmentioning

confidence: 99%

Section: Flat Ballsmentioning

confidence: 99%

Section: Flat Ballsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The analyst's traveling salesman theorem in graph inverse limits

David¹,

Schul²

2017

Ann. Acad. Sci. Fenn. Math.

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Effective Reifenberg theorems in Hilbert and Banach spaces

2018

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A famous theorem by Reifenberg states that closed subsets of R n that look sufficiently close to k-dimensional at all scales are actually C 0,γ equivalent to k-dimensional subspaces. Since then a variety of generalizations have entered the literature. For a general measure µ in R n , one may introduce the k-dimensional Jone's β k -numbers of the measure, where β k (x, r) quantifies on a given ball B r (x) how closely the support of the measure is to living inside a k-dimensional subspace. Recently, it has been proven that if these β-numbers satisfy the uniform summability estimate 1 45 7.2. Construction 47 7.3. Item 2: Graphicality 48 7.4. Item 3: bi-Lipschitz estimates 50 7.6. Item 4: Ball control 52 7.7. Item 5: Radius control 52 7.8. Item 6: Packing control 53 7.9. Item 7: Covering control 53 7.10. Finishing the proof of Lemma 6.3 54 8. Corollaries 54 8.1. Rectifiability 56 8.2. Proof of Proposition 2.9 59 References 63 Remark 2.2. Note that by standard measure theory arguments, a finite Borel measure on a metric space is Borel-regular, see [Par05, theorem II, 1.2, pag 27].Recall from (1.7) the modulus of smoothness ρ X (t) and the smoothness power α for a Banach space X. We will recall the precise definitions of these objects in Section 3.25, here we simply remind the reader that α ∈ [1, 2] and its "best" value α = 2 is achieved by any Hilbert space. For a general Banach space we have α ≥ 1; and for X = L p we have α = min{p, 2} when 1 ≤ p < ∞, and α = 1 when p = ∞.As a corollary, when µ is discrete or has a priori density control, we obtain a measure bound directly. Moreover, we can easily weaken the pointwise assumption (2.1) to an weak-L 1 type assumption. Precisely, we have the following theorem. Corollary 2.3 (Discrete-and Continuous-Reifenberg). Let X be a Banach space, and let µ be a Borel measure with µ(X \ B 1 (0)) = 0. Suppose µ satisfies µ z ∈ B 1 (0) : 2 0

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A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

Hyde¹

2022

Math. Ann.

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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 24 6. Estimating measure 33 7. Bilateral Weak Geometric Lemma 39 Appendix A. Parameterisation in Hilbert space 44 Appendix B. Constants 55 Appendix C. Reduction to Euclidean space 57 References 62

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The traveling salesman problem in the Heisenberg group: Upper bounding curvature

Cited by 31 publications

References 17 publications

The analyst's traveling salesman theorem in graph inverse limits

The analyst's traveling salesman theorem in graph inverse limits

Effective Reifenberg theorems in Hilbert and Banach spaces

A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

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