PLANE WITH $A_{\infty}$ -WEIGHTED METRIC NOT BILIPSCHITZ EMBEDDABLE TO ${\bb R}^n$

Laakso, Tomi

doi:10.1112/s0024609302001200

Cited by 82 publications

(93 citation statements)

References 7 publications

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“…This refines Bourgain's embedding theorem [5], and improves upon previous embeddings of doubling metrics [14]. It is tight for λ X = O(1) [21,22,14], and λ X = n Ω(1) [27]. The question of whether this bound is tight up to a constant factor for the range λ X ∈ {c 1 , .…”

Section: Resultsmentioning

confidence: 62%

Measured descent: a new embedding method for finite metrics

Krauthgamer

Lee

Mendel

et al. 2005

GAFA, Geom. funct. anal.

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We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O( √ α X · log n), where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O( (log λ X ) log n) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, "low-dimensional," improved bounds are achieved.Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in ℓ O(log n) ∞ with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n) 2 ).

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Section: Resultsmentioning

confidence: 62%

Measured descent: a new embedding method for finite metrics

Krauthgamer

Lee

Mendel

et al. 2005

GAFA, Geom. funct. anal.

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“…We will here concentrate on QC and QS maps. Concerning the existence of bi-Lipschitz parametrizations, we only briefly note that interesting sufficient conditions and counterexamples have been found both in the 2-dimensional ( [13], [22], [38], [44], [52], [53]) and higherdimensional cases ( [4], [28], [30], [32], [50]). …”

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confidence: 99%

Uniformization of two-dimensional metric surfaces

Rajala

2016

Invent. math.

156

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Abstract. We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.

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“…We also prove that the distance functions of certain fractal subsets of R n give rise to isotropic doubling weights (Proposition 3.6). By virtue of this result, the sets constructed by Semmes [18] and Laakso [16] provide examples of isotropic doubling weights that are not comparable to Df for any δ-monotone, or even quasiconformal, mapping f : R n → R n .…”

Section: Introductionmentioning

confidence: 99%

“…x ∈ R n (1.3) 1 12C 3 I n−1 µ(x) ≤ Df µ (x) ≤ πI n−1 µ(x), where C is the doubling constant of µ. Theorem 1.1 expands the class of weights that are known to be comparable to Jacobians of quasiconformal mappings. The quasiconformal Jacobian problem posed by David and Semmes in [9] asks for a characterization of all such weights (see also [4,6,7,16,18]). The authors of [6] point out that such a characterization would give a good idea of which metric spaces are bi-Lipschitz equivalent to R n .…”

Section: Introductionmentioning

confidence: 99%

Doubling measures, monotonicity, and quasiconformality

Kovalev

Maldonado

2007

Math. Z.

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Abstract. We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.

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PLANE WITH $A_{\infty}$ -WEIGHTED METRIC NOT BILIPSCHITZ EMBEDDABLE TO ${\bb R}^n$

Cited by 82 publications

References 7 publications

Measured descent: a new embedding method for finite metrics

Measured descent: a new embedding method for finite metrics

Uniformization of two-dimensional metric surfaces

Doubling measures, monotonicity, and quasiconformality

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