Discretization and affine approximation in high dimensions (Q375763)

This important article investigates the macroscopic approximation of Lipschitz maps from a finite-dimensional space \(X\) into a super-reflexive space \(Y\) by affine functions. Given such spaces \(X\) and \(Y\), and real numbers \(\epsilon>0\) and \(r>0\), the condition \(r_{X,Y}(\epsilon)\geq r\) means that every \(Y\)-valued 1-Lipschitz function defined on the unit ball of \(X\) is \(\epsilon\)-close, after renormalization, to an affine function on some ball of radius at least \(r\). The known fact that \(r_{X,Y}(\epsilon)>0\) for all \(\epsilon>0\) was previously obtained by contradiction, through an ultrapower argument which provided no explicit lower bound. The main result of this article is such an explicit lower bound, with a function whose parameters are the dimension \(n\) of \(X\) and the exponent of the modulus of uniform convexity of an equivalent norm on \(Y\). Examples are given which show that this estimate is essentially sharp when \(n=1\). The relation with Bourgain's discretization theorem, which relates the distortion of the Lipschitz embeddings of \(X\) into \(Y\) with the distortion of subnets of \(X\), is established and the value of the lower bound for higher dimensions is shown to match the best-known bounds in Bourgain's theorem for super-reflexive targets. The proofs use very elaborate refinements of the classical mid-point technique which goes back to Enflo's early works.