Differentiability of Lipschitz Functions in Lebesgue Null Sets

TL;DR

This paper demonstrates that in higher dimensions, there are Lebesgue null sets containing points of differentiability for all Lipschitz functions from R^n to R^{n-1}, advancing the understanding of Rademacher's theorem's converse.

Contribution

It proves the existence of specific null sets in R^n that contain differentiability points for all Lipschitz functions, completing the characterization of when Rademacher's theorem's converse holds.

Findings

Existence of null sets with differentiability points for all Lipschitz functions in R^n

Completion of the characterization of Rademacher theorem's converse in higher dimensions

Use of avoidance of sigma-porous sets in the proof

Abstract

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.