Cone unrectifiable sets and non-differentiability of Lipschitz functions

TL;DR

This paper establishes conditions under which certain sets in Euclidean space can be contained in the non-differentiability points of Lipschitz functions, advancing understanding of differentiability and non-differentiability sets.

Contribution

It introduces new sufficient conditions for sets to be non-universal differentiability sets, connecting geometric properties with differentiability behavior of Lipschitz functions.

Findings

Sets satisfying these conditions can be contained in non-differentiability sets of Lipschitz functions.

For every Lebesgue null set, there exists a Lipschitz map non-differentiable on that set.

The results relate geometric set properties to differentiability phenomena in higher dimensions.

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, Cs\"ornyei and Preiss, which eventually led to the result of Jones and Cs\"ornyei 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.