标题： 关于随机形式的函数等周不等式 显示英文标题

标题： On stochastic forms of functional isoperimetric inequalities

摘要： 我们提出了几种功能性等周不等式的概率解释，这些不等式属于$p$-凹函数类，这是基于 P. Pivovarov 和 J. Rebollo-Bueno 引入的此类函数的随机模型。 首先，我们建立了一个经典均质积分功能扩展的随机等周不等式，该不等式在这一随机设置中作为特殊情况给出了一个 Sobolev 型不等式。 受此启发，我们进一步证明了当处理$p$-凹函数的这些随机模型时，Zhang 的仿射 Sobolev 不等式在期望下成立。 最后，我们确认我们的结果恢复了它们的几何类比和确定性对应结果。 作为后者的后果，我们在凸测度的背景下，建立了限制于$p$-凹函数的 Zhang 仿射 Sobolev 不等式的推广。 显示英文摘要

摘要： We present a probabilistic interpretation of several functional isoperimetric inequalities within the class of $p$-concave functions, building on random models for such functions introduced by P. Pivovarov and J. Rebollo-Bueno. First, we establish a stochastic isoperimetric inequality for a functional extension of the classical quermassintegrals, which yields a Sobolev-type inequality in this random setting as a particular case. Motivated by the latter, we further show that Zhang's affine Sobolev inequality holds in expectation when dealing with these random models of $p$-concave functions. Finally, we confirm that our results recover both their geometric analogues and deterministic counterparts. As a consequence of the latter, we establish a generalization of Zhang's affine Sobolev inequality restricted to $p$-concave functions in the context of convex measures.