Transport proofs of weighted Poincar\'e inequalities for log-concave distributions

TL;DR

This paper uses optimal transport methods to establish weighted Poincaré inequalities for log-concave distributions, extending previous results and confirming the variance conjecture for log-concave martingale increments.

Contribution

It introduces transport-based proofs for weighted Poincaré inequalities and verifies the variance conjecture for a new class of log-concave martingale increments.

Findings

Weighted Poincaré inequalities for log-concave vectors established

Variance conjecture proven for log-concave martingale increments

Extension of symmetry-based results to broader log-concave distributions

Abstract

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.