Metric differentiation, monotonicity and maps to L 1

Cheeger, Jeff; Kleiner, Bruce

doi:10.1007/s00222-010-0264-9

Cited by 38 publications

(112 citation statements)

References 28 publications

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“…Section 1.8 of [10] describes an alternate proof of of the non-embeddability of the Heisenberg group into L 1 which uses metric differentiation in the sense of [23,32], and a classification of monotone subsets of the Heisenberg group. This is carried out in full detail in [13].…”

Section: Results and Techniquesmentioning

confidence: 99%

Coarse Differentiation and Multi-flows in Planar Graphs

Lee

Raghavendra

2009

Discrete Comput Geom

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We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound [8] for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair.This also improves the largest known gap for planar graphs from 3 2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the coarse differentiation method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into L 1 .

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Section: Results and Techniquesmentioning

confidence: 99%

Coarse Differentiation and Multi-flows in Planar Graphs

Lee

Raghavendra

2009

Discrete Comput Geom

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“…The main references for this section are [11,12,14,15]. Then we explain the relation between these and Lipschitz maps to L 1 .…”

Section: Tangent Cones Tangent Spaces and Bi-lipschitz Nonembeddingmentioning

confidence: 99%

“…In these cases, monotone sets are half-spaces that might be empty or, at an interior point (density point), might be the whole space; for the case of R n , see [22,23]; for the case of H, see [27]; see also [12,15]. In these cases, monotone sets are half-spaces that might be empty or, at an interior point (density point), might be the whole space; for the case of R n , see [22,23]; for the case of H, see [27]; see also [12,15].…”

Section: Tangent Cones Tangent Spaces and Bi-lipschitz Nonembeddingmentioning

confidence: 99%

Quantitative Differentiation: A General Formulation

Cheeger

2012

Comm Pure Appl Math

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Let dx denote Lebesgue measure on R n . With respect to the measure C D r 1 dr dx on the collection of balls B r .x/ R n , the subcollection of balls B r .x/ B 1 .0/ has infinite measure. Let f W B 1 .0/ ! R have bounded gradient, jrf j Ä 1. For any B r .x/ B 1 .0/ there is a natural scale-invariant quantity that measures the deviation of f j B r .x/ from being an affine linear function. The most basic case of quantitative differentiation (due to Peter Jones) asserts that for all > 0, the measure of the collection of balls on which the deviation from linearity is is finite and controlled by , independent of the particular function f . The purpose of this paper is to explain the sense in which this model case is actually a particular instance of a general phenomenon that is present in many different geometric/analytic contexts. Essentially, in each case that fits the framework, to prove the relevant quantitative differentiation theorem, it suffices to verify that the family of relative defects is monotone and coercive. We indicate one recent application to theoretical computer science and others to partial regularity theory in geometric analysis and nonlinear PDEs.

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“…an RNP Banach space) (see for instance [CK09]), or when B = L 1 (see for instance [CK10a,CK10b,CKN11,CK13]). In connection with embeddings into RNP-Banach spaces, Cheeger and Kleiner [CK09] showed that if (X, µ) is a PI-space the fibres of T X are spanned by "tangent vectors" to Lipschitz curves.…”

Section: Introductionmentioning

confidence: 99%