Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
1
The categories of this paper are: cs