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.

This talk is included in these lists:

• Analysis Seminar
• Analysis seminar
• R17/18
• School of Mathematics Events

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