Discretization and affine approximation in high dimensions

TL;DR

This paper improves bounds on how well vector-valued Lipschitz functions can be approximated by affine maps in high-dimensional normed spaces, offering a new perspective on Bourgain's discretization theorem.

Contribution

It provides sharper lower estimates for affine approximability and introduces a novel approach to Bourgain's discretization theorem for superreflexive spaces.

Findings

Enhanced lower bounds for affine approximability in finite-dimensional normed spaces

A new approach to Bourgain's discretization theorem for superreflexive targets

Completes previous work by Bates et al. on the topic

Abstract

Lower estimates are obtained for the macroscopic scale of affine approximability of vector-valued Lipschitz functions on finite dimensional normed spaces, completing the work of Bates, Johnson, Lindenstrass, Preiss and Schechtman. This yields a new approach to Bourgain's discretization theorem for superreflexive targets.