Heat flow and quantitative differentiation

For every Banach space $(Y,\mathrm{\parallel }\cdot {\mathrm{\parallel }}_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\in (0,\mathrm{\infty })$ with the following property. Suppose that $n\in \mathbb{N}$ and that $X$ is an $n$-dimensional normed space with unit ball $B_X$. Then for every $1$-Lipschitz function $f:B_X\to Y$ and for every $\epsilon \in (0,1\mathrm{/}2]$ there exists a radius $r\ge \exp (-1\mathrm{/}{\epsilon }^{cn})$, a point $x\in B_X$ with $x+rB_X\subset B_X$,andanaffinemapping$\Lambda:X\to Y$suchthat$|f(y)-\Lambda(y)|_Y\le \varepsilon r$forevery$y\in x+rB_X$.