Heat flow and quantitative differentiation

Abstract: For every Banach space $(Y,|\cdot|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\in (0,\infty)$ 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 $\varepsilon\in (0,1/2]$ there exists a radius $r\ge\exp(-1/\varepsilon^{cn})$, a point $x\in B_X$ with $x+rB_X\subset B_X$, and an affine mapping $\Lambda:X\to Y$ such that $|f(y)-\Lambda(y)|_Y\le \varepsilon r$ for every $y\in x+rB_X$.