Quantitative affine approximation for UMD targets

Abstract

It is shown here that if $(Y,\Vert \cdot {\Vert }_Y)$ 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 $\epsilon \in (0,1/2]$, if $(X,\Vert \cdot {\Vert }_X)$ 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 $B^{\ast }=y+\rho B_X\subseteq B_X$ of radius $\rho \ge \exp (-(1/\epsilon {)}^{cn})$ such that $\Vert f(x)-\Lambda (x){\Vert }_Y\le \epsilon \rho$ for all $x\in B^{\ast }$. 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.

arXiv commentThis new version of the article has been reformatted using the Discrete Analysis style, but is otherwise identical to the previous version