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Cone unrectifiable sets and non-differentiability of real-valued Lipschitz functions

Connections for Women: geometry and probability in high dimensions August 17, 2017 - August 18, 2017

August 18, 2017 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Olga Maleva (University of Birmingham)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



There are subsets N of R^n for which one can find a real-valued Lipschitz function f defined on the whole of R^n but non-differentiable 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 non-differentiable 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. 

29292?type=thumb Malvea 92.6 KB application/pdf Download
29350?type=thumb Malvea notes 653 KB application/pdf Download
Video/Audio Files


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