Rigidity and stability of Caffarelli's log-concave perturbation theorem

TL;DR

This paper investigates the rigidity and stability of Caffarelli's log-concave perturbation theorem, showing that measures close to Gaussian in Poincaré constant nearly decompose into Gaussian factors, enhancing understanding of measure structure.

Contribution

The paper provides new rigidity and stability results for Caffarelli's theorem, linking Poincaré constants of log-concave measures to Gaussian factorization.

Findings

Measures with Poincaré constants close to Gaussian nearly split off a Gaussian factor.

Rigidity results characterize when measures are close to Gaussian.

Stability results quantify how small perturbations affect measure structure.

Abstract

In this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almost the same Poincar\'e constant as the Gaussian measure, then it almost splits off a Gaussian factor.