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Anisotropic Preiss density

Max Goering, Max Planck Institute
Tuesday, February 14, 2023 - 1:30pm to 3:30pm
PDL C-401

In the 1920s Besicovitch asked the question: What can one say about the structure of sets $E$ in the plane, with the property that $\lim_{r \downarrow 0} \frac{ \mathcal{H}^{1}(B(x,r) \cap E)}{2r}= 1$ for almost every $x \in E$? In 1987 Preiss gave a complete answer to Besicovitch's density question in groundbreaking work relying on his introduction of the notion of tangent measures.   In the first half of this talk, I will introduce tangent measures and some of their properties that make them widely applicable. In the second half I will briefly introduce several previous results relying on techniques from Preiss' paper and then discuss how, in work-in-progress with Tatiana Toro and Bobby Wilson, we recover anisotropic analogs of several of these results.

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