Quantitative improvements of functional inequalities under concavity properties
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916982421454848 |
|---|---|
| author | Malliaris, Andreas Sola, Francisco Marín |
| author_facet | Malliaris, Andreas Sola, Francisco Marín |
| contents | A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, where $f$ is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and Koldobsky established a quantitative improvement of Hensley's bound. In this work, we complement their result in several directions. First, we prove the corresponding upper bound inequality for $s$-concave functions with $s\geq 0$. Second, we present a generalization of Barthe and Koldobsky's result for functionals of the form $\int_\mathbb{R} Nf\,\mathrm{d}μ$, where $N$ is a convex, even function and $μ$ belongs to a suitable class of positive Borel measures. As a consequence of the employed methods, we obtain quantitative refinements of classical inequalities for $p$-norms and for the entropy of log-concave functions. Finally, we discuss both geometric consequences and probabilistic interpretations of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative improvements of functional inequalities under concavity properties Malliaris, Andreas Sola, Francisco Marín Functional Analysis Metric Geometry A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, where $f$ is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and Koldobsky established a quantitative improvement of Hensley's bound. In this work, we complement their result in several directions. First, we prove the corresponding upper bound inequality for $s$-concave functions with $s\geq 0$. Second, we present a generalization of Barthe and Koldobsky's result for functionals of the form $\int_\mathbb{R} Nf\,\mathrm{d}μ$, where $N$ is a convex, even function and $μ$ belongs to a suitable class of positive Borel measures. As a consequence of the employed methods, we obtain quantitative refinements of classical inequalities for $p$-norms and for the entropy of log-concave functions. Finally, we discuss both geometric consequences and probabilistic interpretations of our results. |
| title | Quantitative improvements of functional inequalities under concavity properties |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2510.00645 |