Quantitative Affine Approximation for Umd Targets

It is shown here that if (Y, ‖ · ‖Y ) is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists c = c(Y ) ∈ (0,∞) with the following property. For every n ∈ N and ε ∈ (0, 1/2], if (X, ‖·‖X) is an n-dimensional normed space with unit ball BX and f : BX → Y is a 1-Lipschitz function then there exists an affine mapping Λ : X → Y and a sub-ball B∗ = y + ρBX ⊆ BX of radius ρ > exp(−(1/ε)) such that ‖f(x) − Λ(x)‖Y 6 ερ for all x ∈ B∗. This estimate on the macroscopic scale of affine approximability of vector-valued Lipschitz functions is an asymptotic improvement (as n → ∞) over the best previously known bound even when X is R equipped with the Euclidean norm and Y is a Hilbert space.