On stochastic forms of functional isoperimetric inequalities
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908960867483648 |
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| author | Sola, Francisco Marín |
| author_facet | Sola, Francisco Marín |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04108 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On stochastic forms of functional isoperimetric inequalities Sola, Francisco Marín Functional Analysis Metric Geometry Probability 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. |
| title | On stochastic forms of functional isoperimetric inequalities |
| topic | Functional Analysis Metric Geometry Probability |
| url | https://arxiv.org/abs/2509.04108 |