Sharp lower bound for the Monge-Ampère torsion on convex sets
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908774407602176 |
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| author | Salerno, Francesco |
| author_facet | Salerno, Francesco |
| contents | The \emph{Monge-Ampère} torsion deficit of an open, bounded convex set $Ω\subset\R^n$ of class $C^2$ is the normalized gap between the value of the torsion functional evaluated on $Ω$ and its value on the ball with the same $(n-1)$-quermassintegral as $Ω$. Using the technique of the \emph{shape derivative}, we prove that the ratio between this deficit and to a geometric deficit arising from the \emph{Alexandrov-Fenchel inequality}, for any given family of open, bounded convex sets of $\R^n$ ($n\geq2$) of class $C^2$, smoothly converging to a ball, is bounded from below by a dimensional constant. We also show that this ratio is always bounded from above by a constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12915 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp lower bound for the Monge-Ampère torsion on convex sets Salerno, Francesco Analysis of PDEs 35B35, 35J96, 52A40 The \emph{Monge-Ampère} torsion deficit of an open, bounded convex set $Ω\subset\R^n$ of class $C^2$ is the normalized gap between the value of the torsion functional evaluated on $Ω$ and its value on the ball with the same $(n-1)$-quermassintegral as $Ω$. Using the technique of the \emph{shape derivative}, we prove that the ratio between this deficit and to a geometric deficit arising from the \emph{Alexandrov-Fenchel inequality}, for any given family of open, bounded convex sets of $\R^n$ ($n\geq2$) of class $C^2$, smoothly converging to a ball, is bounded from below by a dimensional constant. We also show that this ratio is always bounded from above by a constant. |
| title | Sharp lower bound for the Monge-Ampère torsion on convex sets |
| topic | Analysis of PDEs 35B35, 35J96, 52A40 |
| url | https://arxiv.org/abs/2601.12915 |