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

TL;DR

This paper introduces a novel infinitesimal form of the Prékopa-Leindler inequality with a multiplicative structure, leading to new insights and applications in geometric and analytic inequalities, including stability estimates and the Brunn-Minkowski conjecture.

Contribution

It develops an alternative to the classical $L_2$ method by exploiting a multiplicative structure derived from a concavity principle, advancing the study of inequalities.

Findings

Strengthened weighted boundary Poincaré inequality

Derived stability estimate for weighted Poincaré inequality

Obtained new reformulations and partial results for the Brunn-Minkowski conjecture

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.