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

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arxiv 2501.04656 v1 pith:UIE5ZVTT submitted 2025-01-08 math.FA math.APmath.COmath.MGmath.PR

Alessio Figalli , Peter van Hintum , Marius Tiba This is my paper · ORCID
classification math.FAmath.APmath.COmath.MGmath.PR
keywords inequalitystabilityekopa-leindlersharpborell-brascamp-liebinequalitiesquantitativebrunn-minkowski
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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.

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Cited by 3 Pith papers

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  1. On Ball's conjectured Santal\'o type inequality

    math.MG 2026-02 conditional novelty 8.0 of 10

    Ball's 1986 Santaló-type inequality for symmetric convex bodies is proved in full generality, with equality only for ellipsoids, plus an asymptotically optimal stability estimate.

  2. Quantitative stability for the Brascamp-Lieb inequality and moment measures

    math.FA 2025-11 conditional novelty 7.0 of 10

    For any convex potential, the L1 distance from a function to the Brascamp-Lieb optimizer manifold is controlled by the square root of its deficit, with a dimension-only constant independent of the potential.

  3. Tight Stability Estimates Near the Simplex and Improved Bounds for the Diameter of the Banach-Mazur Compactum in Fixed Dimensions

    math.MG 2026-07 accept novelty 6.0 of 10

    If s(K) ≥ n−ε then d_BM(K, simplex) ≤ 1+ε+ε²/(2(1−ε)), optimally linear in ε, with applications to ball-distance stability and Banach–Mazur diameter bounds.

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