Cheeger's differentiation theorem via the multilinear Kakeya inequality

Abstract

Suppose that $(X,d,\mu )$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f:X\to \mathbb{R}$, $\mathrm{Lip}(f,\cdot )$ 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 Poincaré 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 $n$.

v2: adjust introduction, incorporate referee's comments

References in corpus (4)

• A short proof of the multilinear Kakeya inequality
• Derivations and Alberti representations
• Characterizations of rectifiable metric measure spaces
• Differentiability and Poincaré-type inequalities in metric measure spaces

Cited by in corpus (1)

• Quantitative Alberti representations in spaces of bounded geometry