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Cone unrectifiable sets and non-differentiability of Lipschitz functions

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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.

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Author information

Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

    Olga Maleva

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    David Preiss

Authors
  1. Olga Maleva
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  2. David Preiss
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Corresponding author

Correspondence to Olga Maleva.

Additional information

The research leading to these results has received funding from the European Research Council / ERC Grant Agreement n. 291497. The first-named author also acknowledges the support of the EPSRC grant EP/N027531/1 and of the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

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Maleva, O., Preiss, D. Cone unrectifiable sets and non-differentiability of Lipschitz functions. Isr. J. Math. 232, 75–108 (2019). https://doi.org/10.1007/s11856-019-1863-9

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  • DOI: https://doi.org/10.1007/s11856-019-1863-9

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