Transport proofs of weighted Poincar\'e inequalities for log-concave   distributions
  We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave random vectors with symmetries. In addition, we prove that the variance conjecture is true for increments of log-concave martingales. 
Dario Cordero-Erausquin (IMJ), Nathael Gozlan (LAMA)
math.PR math.FA