Transport proofs of weighted Poincare inequalities for, log-concave distributions

We prove, using optimal transport tools, weighted Poincare 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.