A new infinitesimal form of the Prékopa-Leindler inequality with multiplicative structure and applications

By differentiating a concavity principle arising from the Prékopa-Leindler inequality, we obtain a statement simultaneously strengthening the weighted boundary Poincaré 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 Poincaré inequality and to investigate the dimensional Brunn-Minkowski conjecture. In particular, in the latter setting, we obtain new reformulations together with several partial results.