Cone unrectifiable sets and nondifferentiability of realvalued Lipschitz functions
Connections for Women: geometry and probability in high dimensions August 17, 2017  August 18, 2017
Location: SLMath: Eisenbud Auditorium
4Maleva
There are subsets N of R^n for which one can find a realvalued Lipschitz function f defined on the whole of R^n but nondifferentiable at every point of N. Of course, by the Rademacher theorem any such set N is Lebesgue null, however, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of R^n gives rise to such a Lipschitz function f.
In this talk, I explain that a sufficient condition on a set N for such f to exist is being locally unrectifiable with respect to curves in a cone of directions. In particular, every purely unrectifiable set U possesses a Lipschitz function nondifferentiable on U in the strongest possible sense. I also give an example of a universal differentiability set unrectifiable with respect to a fixed cone of directions, showing that one cannot relax the conditions.
This is a joint work with D. Preiss.
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