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

Cheeger's differentiation theorem via the multilinear Kakeya inequality