Michael Dymond

Differentiability of typical Lipschitz mappings

Rademacher's theorem states that Lipschitz mappings between Euclidean spaces are differentiable almost everywhere with respect to the Lebesgue measure. So for any such Lipschitz mapping, the set of its non-differentiability points has Lebesgue measure zero. We will investigate an even smaller class of sets, namely those in which not only one but many Lipschitz mappings fail to find a point of differentiability. Conversely, we also address the question of what type of "largeness" of a set is required in order to guarantee that most Lipschitz functions have points of differentiability inside of it. We will discuss differentiability and non-differentiability sets of the typical Lipschitz mapping, where typical is understood in the sense of the Baire Category theorem. A geometric characterisation of these sets inside the class of analytic sets will be presented. This is based on joint work with Olga Maleva (Birmingham).