Olga Maleva: Differentiability of typical Lipschitz functions

Abstract: The classical Rademacher Theorem guarantees that every Lipschitz function between finitedimensional spaces is differentiable almost everywhere. There are, however, null subsets S of R^n (with n greater than 1) with the property that every Lipschitz function on R^n has points of differentiability in S; one says that S is a universal differentiability set (UDS).
It turns out that some sets T which are not UDS still have the property that a typical Lipschitz function (understood in the sense of Baire category) has points of full differentiability in T. We characterize such sets completely in the language of Geometric Measure Theory: these are exactly 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, even directionally, at each point. Surprisingly though, no matter how good the set T is, we show that a typical 1-Lipschitz function is non-differentiable at a typical point of T in a very strong sense. This is a joint work with Michael Dymond.