Rigidity and stability of Caffarelli's log-concave perturbation theorem
  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. 
Guido De Philippis, Alessio Figalli
math.AP