Sharp Quantitative Stability for the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb Inequalities

TL;DR

This paper establishes the sharpest possible quantitative stability results for the Borell-Brascamp-Lieb inequality, thereby resolving the long-standing conjecture for the Prékopa-Leindler inequality, and builds on recent advances in the Brunn-Minkowski inequality.

Contribution

It provides a unified, optimal stability framework for Borell-Brascamp-Lieb and Prékopa-Leindler inequalities, solving a major open problem in the field.

Findings

Optimal quantitative stability for Borell-Brascamp-Lieb established

Sharp stability for Prékopa-Leindler inequality proved as a special case

Unified approach based on recent Brunn-Minkowski stability results

Abstract

The Borell-Brascamp-Lieb inequality is a classical extension of the Pr\'ekopa-Leindler inequality, which in turn is a functional counterpart of the Brunn-Minkowski inequality. The stability of these inequalities has received significant attention in recent years. Despite substantial progress in the geometric setting, a sharp quantitative stability result for the Pr\'ekopa-Leindler inequality has remained elusive, even in the special case of log-concave functions. In this work, we provide a unified and definitive stability framework for these foundational inequalities. By establishing the optimal quantitative stability for the Borell-Brascamp-Lieb inequality in full generality, we resolve the conjectured sharp stability for the Pr\'ekopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors.