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Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities

Alessio Figalli, Peter van Hintum, Marius Tiba

TL;DR

This work delivers a sharp, quantitative stability theory for the Borell-Brascamp-Lieb inequality valid for all $n\in\mathbb{N}$, $\lambda\in(0,1/2]$, and $p\in(-1/n,\infty)$. By leveraging a novel combination of stability for Brunn-Minkowski, level-set transport, and a hierarchical cone/fiber decomposition, the authors first bound the symmetric difference $\int|f-g|$ up to a translation by $O_{n,\lambda,p}(\sqrt{\delta})$, then in the case $f=g$ construct a $p$-concave function $\ell$ with $\int|f-\ell|$ also controlled at the same rate. The main theorem yields the optimal $\sqrt{\delta}$-stability for BBLI and, as a special case, the sharp stability for Prékopa-Leindler, resolving longstanding conjectures. The results hinge on a refined, transport-based analysis of level sets, a two-dimensional backbone strengthened by a comprehensive $n$-dimensional reduction, and a robust framework (BMStab) for tracking how near-equality in Brunn-Minkowski propagates to functional inequalities. These findings bridge geometric and functional stability theories with sharp exponents, offering precise stability diagnostics for foundational inequalities in convex geometry and analysis.

Abstract

The Borell-Brascamp-Lieb inequality is a classical extension of the Prékopa-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 Prékopa-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 Prékopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors.

Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities

TL;DR

This work delivers a sharp, quantitative stability theory for the Borell-Brascamp-Lieb inequality valid for all , , and . By leveraging a novel combination of stability for Brunn-Minkowski, level-set transport, and a hierarchical cone/fiber decomposition, the authors first bound the symmetric difference up to a translation by , then in the case construct a -concave function with also controlled at the same rate. The main theorem yields the optimal -stability for BBLI and, as a special case, the sharp stability for Prékopa-Leindler, resolving longstanding conjectures. The results hinge on a refined, transport-based analysis of level sets, a two-dimensional backbone strengthened by a comprehensive -dimensional reduction, and a robust framework (BMStab) for tracking how near-equality in Brunn-Minkowski propagates to functional inequalities. These findings bridge geometric and functional stability theories with sharp exponents, offering precise stability diagnostics for foundational inequalities in convex geometry and analysis.

Abstract

The Borell-Brascamp-Lieb inequality is a classical extension of the Prékopa-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 Prékopa-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 Prékopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors.
Paper Structure (41 sections, 43 theorems, 330 equations)

This paper contains 41 sections, 43 theorems, 330 equations.

Table of Contents

  1. Introduction
  2. The Brunn-Minkowski inequality
  3. The Prékopa-Leindler inequality
  4. The Borell-Brascamb-Lieb inequality
  5. Main results
  6. Proof of \ref{['main']}
  7. Structure of the paper
  8. Notation
  9. \ref{['symmetric_diff']}: structure of the proof
  10. Overview of the 1-dimensional proof
  11. Overview of the 2-dimensional proof
  12. Overview of the $n$-dimensional proof
  13. Preliminary results for the proof of \ref{['symmetric_diff']}
  14. Proof of Lemmas
  15. Proof of the reduction
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $|A|=|B|$ and $\lambda \in (0,1/2]$, and assume that $|\lambda A+(1-\lambda)B|\leq (1+\delta)|A|$ for some $\delta>0$ sufficiently small in terms of $n$ and $\lambda$. Then there exists a convex set $K$ so that, up to a translation,

Theorems & Definitions (131)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Borell-Brascamb-Lieb Inequality
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 121 more