The Brunn–Minkowski and Prékopa–Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn–Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of [Formula: see text]-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa–Leindler inequality.

中文翻译：

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对数凹函数的随机 Prékopa-Leindler 不等式

Brunn-Minkowski 和 Prékopa-Leindler 不等式承认受凸性启发的各种证明。然而，前者适用于紧集，后者适用于可积函数，因此凸性似乎没有特殊意义。另一方面，最近表明，专门针对凸集的 Brunn-Minkowski 不等式源于自然相关的随机多面体的局部随机优势。我们表明，对于 [公式：见文本]-凹函数的子类和相关的随机近似，类似的随机优势是 Prékopa-Leindler 不等式的基础。