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arXiv · 1904.00808

Cheeger's differentiation theorem via the multilinear Kakeya inequality

David Bate·Ilmari Kangasniemi·Tuomas Orponen
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Abstract

Suppose that $(X,d,\mu)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincar\'e inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbourhoods of Lipschitz graphs to show that any non-trivial measure with $n$ independent Alberti representations has Hausdorff dimension at least $n$.

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David Bate, Ilmari Kangasniemi, Tuomas Orponen. 2019-04-01. Cheeger's differentiation theorem via the multilinear Kakeya inequality. https://arxiv.org/abs/1904.00808

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