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<p>We present a probabilistic interpretation of several functional isoperimetric inequalities within the class of <i>p</i>-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 <i>p</i>-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 <i>p</i>-concave functions in the context of convex measures.</p>

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On stochastic forms of functional isoperimetric inequalities

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月7日

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  • Francisco Marín Sola

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.