Maleva, Olga and Preiss, David (2019) Cone unrectifiable sets and non-differentiability of Lipschitz functions. Israel Journal of Mathematics, 232 . pp. 75-108. doi:10.1007/s11856-019-1863-9 ISSN 0021-2172.

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Abstract

We provide sufficient conditions for a set E ⊂ ℝn 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 ℝn given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set E in ℝn there is a Lipschitz map f: ℝn → ℝn not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝn 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Functional analysis, Radon measures
Journal or Publication Title:	Israel Journal of Mathematics
Publisher:	Magnes Press
ISSN:	0021-2172
Official Date:	August 2019
Dates:	Date Event August 2019 Published 30 April 2019 Available 8 August 2018 Accepted
Volume:	232
Page Range:	pp. 75-108
DOI:	10.1007/s11856-019-1863-9
Status:	Peer Reviewed
Publication Status:	Published
Reuse Statement (publisher, data, author rights):	This is a post-peer-review, pre-copyedit version of an article published in Israel Journal of Mathematics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11856-019-1863-9
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	15 August 2018
Date of first compliant Open Access:	30 April 2020
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID 291497 European Research Council http://viaf.org/viaf/130022607 EP/N027531/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 DMS-1440140 National Science Foundation http://dx.doi.org/10.13039/100000001

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