A new infinitesimal form of the Pr\'ekopa-Leindler inequality with multiplicative structure and applications

Abstract

By differentiating a concavity principle arising from the Pr\'ekopa-Leindler inequality, we obtain a statement simultaneously strengthening the weighted boundary Poincar\'e inequality and the Brascamp-Lieb variance inequality. The resulting inequality possesses a multiplicative structure, which we exploit to develop an alternative to the (by now classical) $L_2$ method in the study of geometric and analytic inequalities. We apply this approach to derive a stability estimate for the weighted Poincar\'e inequality and to investigate the dimensional Brunn-Minkowski conjecture. In particular, in the latter setting, we obtain new reformulations together with several partial results.