Heat flow and quantitative differentiation

Abstract

For every Banach space $(Y,\Vert \cdot {\Vert }_Y)$ 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 $\epsilon \in (0,1/2]$ there exists a radius $r\ge \exp (-1/{\epsilon }^{cn})$, a point $x\in B_X$ with $x+r{B}_X\subset B_X$, and an affine mapping $\Lambda :X\to Y$ such that $\Vert f(y)-\Lambda (y){\Vert }_Y\le \epsilon r$ for every $y\in x+r{B}_X$.