A Berry-Esseen type inequality for convex bodies with an unconditional basis

We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal transportation of measures.

💡 Research Summary

The paper studies the central limit theorem (CLT) for high‑dimensional random vectors whose distribution is log‑concave and invariant under coordinate sign changes (the so‑called unconditional property). The main setting is a random vector X=(X₁,…,Xₙ) either uniformly distributed in a convex body K⊂ℝⁿ or, more generally, having a density f=e^{-H} with H convex. The authors assume isotropic normalization, i.e. 𝔼X_i²=1 for every i, and the unconditional symmetry f(x)=f(±x₁,…,±xₙ). Under these hypotheses they obtain two sharp quantitative results.

1. Thin‑shell estimate. They prove that the variance of the squared Euclidean norm satisfies
   Var(|X|²)=𝔼