Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Romney, Matthew; Vellis, Vyron

doi:10.4310/mrl.2017.v24.n4.a11

Cited by 11 publications

(16 citation statements)

References 24 publications

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“…), set s 0 = − log 4/ log p 0 , and fix s ∈ [1, s 0 ). It is well-known that there exists a (1/s 0 )-bi-Hölder homeomorphism Φ : [0, 1] → S p ; e.g., see [BH04,RV17].…”

Section: Examplesmentioning

confidence: 99%

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Badger

Naples

Vellis

2019

Advances in Mathematics

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We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a (1/s)-Hölder continuous map f : [0, 1] → l 2 , with s > 1. Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R N or l 2 in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higherdimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves f k ([0, 1]). We then find conditions in terms of tube approximations that ensure the approximating curves converge to a Hölder curve. As an application to the geometry of measures, we identify conditions that guarantee fractional rectifiability of pointwise doubling measures in R N .

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“…), set s 0 = − log 4/ log p 0 , and fix s ∈ [1, s 0 ). It is well-known that there exists a (1/s 0 )-bi-Hölder homeomorphism Φ : [0, 1] → S p ; e.g., see [BH04,RV17].…”

Section: Examplesmentioning

confidence: 99%

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Badger

Naples

Vellis

2019

Advances in Mathematics

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“…These facts, combined with a point of density argument, prevent any bilipschitz embedding of G 2 α into R α . By generalizing the construction in [27], Romney and Vellis [20] recently showed that G 2 α can be bilipschitzly embedded into R 2+ α for each α > 0, where α is the integer part of α.…”

Section: Discussionmentioning

confidence: 99%

“…The existence of bilipschitz embedding of the Grushin plane G 2 α into R 3 has been proved in [27] and [20] for 0 < α 1 < 2; see the discussion in Section 1.1.…”

Section: Bilipschitz Parametrization and Embedding Of Grushin Spacesmentioning

confidence: 99%

Geometry of Grushin spaces

Wu¹

2015

Illinois J. Math.

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We compare the Grushin geometry to Euclidean geometry, through quasisymmetric parametrization, bilipschitz parametrization and bilipschitz embedding, highlighting the role of the exponents and the fractal nature of the singular hyperplanes in Grushin geometry. ∂x 2 j associated to the vector fields studied in this note have been the focus in [6], [12], [13], [17], and [18]. Recent solutions of isoperimetric problems on Grushin planes by Monti and Date: February 28, 2015.

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“…Let β = 1−ǫ and let Y = {(0, v) : v ∈ R}. There exists a global bi-Lipschitz map Γ : [20,Cor. 1.3] in terms of our new notation for Grushin-type surfaces.…”

Section: For the Last Inequality Notice Thatmentioning

confidence: 99%

“…This shows that Γ is bi-Lipschitz as a mapping from the ( Y , β + β − ββ)-Grushin plane to the (Y, β)-Grushin plane. The ( Y , β + β − ββ)-Grushin plane is isometric (up to rescaling) to the usual [20].…”

Section: For the Last Inequality Notice Thatmentioning

confidence: 99%

Conformal Grushin spaces

Romney

2016

Conform. Geom. Dyn.

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Abstract. We introduce a class of metrics on R n generalizing the classical Grushin plane. These are length metrics defined by the line element ds = dE(·, Y ) −β dsE for a closed nonempty subset Y ⊂ R n and β ∈ [0, 1). We prove that, assuming a Hölder condition on the metric, these spaces are quasisymmetrically equivalent to R n and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional case, we give another proof of bi-Lipschitz embeddability based on growth bounds on sectional curvature.

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Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces

Abstract: We show that, for all α ≥ 0, the generalized Grushin plane Gα is bi-Lipschitz homeomorphic to a 2-dimensional quasiplane in the Euclidean space R [α]+2 , where [α] is the integer part of α. The target dimension is sharp. This generalizes a recent result of Wu [22].

Cited by 11 publications

References 24 publications

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

Geometry of Grushin spaces

Conformal Grushin spaces

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