Event Details

Dichotomy of subsets of R^n via typical differentiability

The classical Rademacher Theorem guarantees that every set of positive Lebesgue measure in a finite-dimensional Euclidean space contains points of differentiability of every Lipschitz (real-valued) function defined on the whole space.

A major direction in geometric measure theory research of the last two decades was to explore to what extent this is true for Lebesgue null subsets offinite-dimensional spaces. First, Zahorski showed in the 1940s that for any null subset N of R there is a Lipschitz function defined on R nowhere differentiable in N. In contrast, Preiss proved in 1990 that every finite-dimensional space of dimension at least 2 has Lebesgue null subsets S such that every Lipschitz function on the whole space has points of differentiability in S. Sets with the latter property are called universal differentiability sets (UDS), and examples with additional important features (such as closed and of Minkowski dimension 1) of such sets were constructed in joint works with Dore and with Dymond.

But if there exists a Lipschitz function nowhere differentiable on a given set N, one naturally wonders what happens with a typical (in the sense of Baire category) Lipschitz function. In this talk, I will present a recent joint work with Dymond, where we show how a question of differentiability of a typical Lipschitz function inside a given analytic subset of a finite-dimensional space is settled. Namely, we give a complete characterisation of the sets in which a typical 1-Lipschitz function has points of differentiability: these are the sets which cannot be covered by an F-sigma 1-purely unrectifiable set. We also show that for all remaining sets a typical 1-Lipschitz function is nowhere differentiable, not even directionally, at each point. The proof involves topological Banach-Mazur game.