Differentiability of Lipschitz Functions inside Negligible Sets

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• Michael Dymond (Birmingham)
• Wednesday 20 February 2013, 16:00-17:00
• R17/18.

If you have a question about this talk, please contact Neal Bez.

Rademacher’s Theorem states that Lipschitz functions on Euclidean spaces are differentiable almost everywhere with respect to the Lebesgue measure. Moreover, in $\mathbb{R}$, any set $N$ of Lebesgue measure zero admits a Lipschitz function on $\mathbb{R}$, nowhere differentiable on $N$. However, the situation is vastly different in Euclidean spaces of dimension higher than one. In 1990, Preiss gave an example of a Lebesgue null subset of the plane containing a differentiability point of every Lipschitz function on $\mathbb{R}²$. This set contains every line segment between points of $\mathbb{R}²$ with rational co-ordinates. In some sense, Preiss’ set is still rather large; its closure is the whole of $\mathbb{R}²$. Thus, it is natural to ask whether we can force differentiability of every Lipschitz function inside much smaller sets. In recent work of Maleva and Dor\’{e}, the existence of a compact universal differentiability set with Hausdorff dimension one is verified. The Hausdorff dimension is bounded above by the (upper and lower) Minkowski dimensions, and the question of whether there exists a universal differentiability set with upper or lower Minkowksi dimension one has remained open. We discuss some new results including a construction of a universal differentiability set having both upper and lower Minkowski dimension one, and a general property of universal differentiability sets. We also discuss possible directions for future research in this area.

This talk is part of the Analysis seminar series.

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• Analysis seminar
• Analysis seminar
• R17/18
• School of Mathematics events

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