Quantitative affine approximation for UMD targets

It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the following property. For every $n\in \mathbb{N}$ and $\varepsilon\in (0,1/2]$, if $(X,\|\cdot\|_X)$ is an $n$-dimensional normed space with unit ball $B_X$ and $f:B_X\to Y$ is a $1$-Lipschitz function then there exists an affine mapping $\Lambda:X\to Y$ and a sub-ball $B^*=y+\rho B_X\subseteq B_X$ of radius $\rho\ge \exp(-(1/\varepsilon)^{cn})$ such that $\|f(x)-\Lambda(x)\|_Y\le \varepsilon \rho$ for all $x\in B^*$. This estimate on the macroscopic scale of affine approximability of vector-valued Lipschitz functions is an asymptotic improvement (as $n\to \infty$) over the best previously known bound even when $X$ is $\mathbb{R}^n$ equipped with the Euclidean norm and $Y$ is a Hilbert space.