Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Salerno, Francesco
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7125264232
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