Cone unrectifiable sets and non-differentiability of Lipschitz functions

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Cone unrectifiable sets and non-differentiability of Lipschitz functions. / Maleva, Olga; Preiss, David.

In: Israel Journal of Mathematics, Vol. 232, No. 1, 08.2019, p. 75-108.

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@article{42ec211be2bc4c2e9944841b31a637a0,
title = "Cone unrectifiable sets and non-differentiability of Lipschitz functions",
abstract = "We provide sufficient conditions for a set E ⊂ ℝ n to be a non-universal differentiability set, i.e., to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of ℝ n given by Alberti, Cs{\"o}rnyei and Preiss, which eventually led to the result of Jones and Cs{\"o}rnyei that for every Lebesgue null set E in ℝ n there is a Lipschitz map f: ℝ n → ℝ n not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝ n to ℝ there exist Lebesgue null universal differentiability sets. Among other results, we show that the new class of Lebesgue null sets introduced here contains all uniformly purely unrectifiable sets and gives a quantified version of the result about non-differentiability in directions outside the decomposability bundle with respect to a Radon measure. ",
author = "Olga Maleva and David Preiss",
year = "2019",
month = aug,
doi = "10.1007/s11856-019-1863-9",
language = "English",
volume = "232",
pages = "75--108",
journal = "Israel Journal of Mathematics",
issn = "0021-2172",
publisher = "Springer",
number = "1",

}

RIS

TY - JOUR

T1 - Cone unrectifiable sets and non-differentiability of Lipschitz functions

AU - Maleva, Olga

AU - Preiss, David

PY - 2019/8

Y1 - 2019/8

N2 - We provide sufficient conditions for a set E ⊂ ℝ n to be a non-universal differentiability set, i.e., to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of ℝ n given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set E in ℝ n there is a Lipschitz map f: ℝ n → ℝ n not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝ n to ℝ there exist Lebesgue null universal differentiability sets. Among other results, we show that the new class of Lebesgue null sets introduced here contains all uniformly purely unrectifiable sets and gives a quantified version of the result about non-differentiability in directions outside the decomposability bundle with respect to a Radon measure.

AB - We provide sufficient conditions for a set E ⊂ ℝ n to be a non-universal differentiability set, i.e., to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of ℝ n given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set E in ℝ n there is a Lipschitz map f: ℝ n → ℝ n not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝ n to ℝ there exist Lebesgue null universal differentiability sets. Among other results, we show that the new class of Lebesgue null sets introduced here contains all uniformly purely unrectifiable sets and gives a quantified version of the result about non-differentiability in directions outside the decomposability bundle with respect to a Radon measure.

UR - http://www.scopus.com/inward/record.url?scp=85070413639&partnerID=8YFLogxK

U2 - 10.1007/s11856-019-1863-9

DO - 10.1007/s11856-019-1863-9

M3 - Article

VL - 232

SP - 75

EP - 108

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -