Cone unrectifiable sets and non-differentiability of Lipschitz functions

Abstract

We provide sufficient conditions for a set $E\subset {\mathbb{R}}^n$ 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 ${\mathbb{R}}^n$ given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set $E$ in ${\mathbb{R}}^n$ there is a Lipschitz map $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ not differentiable at any point of $E$, even though for $n>1$ and for Lipschitz functions from ${\mathbb{R}}^n$ to $\mathbb{R}$ there exist Lebesgue null universal differentiability sets.

30 pages