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arXiv · 2501.04656

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

Alessio Figalli·Peter van Hintum·Marius Tiba
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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.

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Alessio Figalli, Peter van Hintum, Marius Tiba. 2025-01-08. Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities. https://arxiv.org/abs/2501.04656

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