Prékopa-Leindler 不等式与 Borell-Brascamp-Lieb 不等式的尖锐定量稳定性
Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities
AI总结：
本文构建统一框架，在完全一般条件下证明 Borell-Brascamp-Lieb 不等式的最优定量稳定性，并据此解决 Prékopa-Leindler 不等式的尖锐稳定性猜想。
摘要：
Borell-Brascamp-Lieb 不等式是 Prékopa-Leindler 不等式的经典推广，而后者又是 Brunn-Minkowski 不等式的泛函对应形式。近年来，这些不等式的稳定性受到广泛关注。尽管在几何情形中已取得实质性进展，但即使在对数凹函数这一特殊情形下，Prékopa-Leindler 不等式的尖锐定量稳定性结果仍难以获得。本文为这些基础性不等式提供了一个统一且确定的稳定性框架。通过在完全一般的条件下建立 Borell-Brascamp-Lieb 不等式的最优定量稳定性，我们作为特例解决了人们猜想的 Prékopa-Leindler 不等式尖锐稳定性问题。我们的方法建立在作者近期获得的 Brunn-Minkowski 不等式尖锐稳定性结果之上。
英文摘要：
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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