Characterizations of countably $n$-rectifiable Radon measures by higher-dimensional Menger curvatures

Goering, Max

doi:10.48550/arxiv.1804.02497

Cited by 3 publications

(4 citation statements)

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“…For related work on rectifiability of absolutely continuous measures and the theory of mass transport, see Tolsa [Tol12], Azzam, David, and Toro [ADT16, ADT17], and Azzam, Tolsa, and Toro [ATT18]. For the connection between rectifability of sets and Menger-type curvatures, see Léger [Lég99], Lerman and Whitehouse [LW11, LW09], Meurer [Meu18], and Goering [Goe18]. For related results about discrete approximation and rectifiability of varifolds, see Buet [Bue15].…”

Section: Theorem 35 (Assorted Characterizations Of Absolutely Continu...mentioning

confidence: 99%

Generalized rectifiability of measures and the identification problem

Badger

2019

Complex Anal Synerg

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One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or Hölder or C 1,α images of Euclidean subspaces, including theorems of Azzam-Tolsa, Badger-Schul, Badger-Vellis, Edelen-Naber-Valtorta, Ghinassi, and Tolsa-Toro.This survey paper is based on a talk at the Northeast

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Section: Theorem 35 (Assorted Characterizations Of Absolutely Continu...mentioning

confidence: 99%

Generalized rectifiability of measures and the identification problem

Badger

2019

Complex Anal Synerg

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“…A description of this work as it stood at the end of the last century can be found can be found in [41]. Newer developments in the theory of rectifiability of absolutely continuous measures include [7,21,22,31,35,36,48,50]. An alternative regularity condition that is usually a priori weaker than upper and lower density bounds is asymptotic control on how much the measure grows when the radius of a ball is doubled.…”

Section: Introductionmentioning

confidence: 99%

Radon measures and Lipschitz graphs

Badger

Naples

2021

Bulletin of London Math Soc

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For all 1⩽m⩽n−1, we investigate the interaction of locally finite measures in Rn with the family of m‐dimensional Lipschitz graphs. For instance, we characterize Radon measures μ, which are carried by Lipschitz graphs in the sense that there exist graphs normalΓ1,normalΓ2,⋯ such that μ(double-struckRn∖⋃1∞normalΓi)=0, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, for example, for the restrictions of m‐dimensional Hausdorff measure Hm to E⊆double-struckRn with 0

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“…Nonetheless, geometric arguments made with non-trivial adaptations from [Lég99] have since been used to characterize uniform rectifiability in all dimensions and codimensions in terms of Menger-type curvatures, [LW09,LW11]. A sufficient condition for rectifiability of sets in terms of higher dimensional Menger-type curvatures appears in [Meu18] and was extended to several characterizations of rectifiable measures under suitable density conditions [Goe18].…”

Section: Introductionmentioning

confidence: 99%

“…The α = 0 case in Theorem I appears in [Goe18]. The case α > 0 is an improvement of a special case of [Kol17] where the lower density assumption is relaxed.…”

Section: Introductionmentioning

confidence: 99%