Cone unrectifiable sets and non-differentiability of Lipschitz functions

Research output: Contribution to journalArticle

Authors

Colleges, School and Institutes

External organisations

  • University of Warwick

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, 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.

Details

Original languageEnglish
Pages (from-to)75-108
Number of pages34
JournalIsrael Journal of Mathematics
Volume232
Issue number1
Early online date30 Apr 2019
Publication statusPublished - Aug 2019

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