Sharp lower bound for the Monge-Ampère torsion on convex sets

TL;DR

This work studies the relationship between the Monge-Ampère torsion deficit $\delta_T$ and the Alexandrov–Fenchel deficit $\delta_{AF}$ for open bounded convex sets. Using shape derivatives, it derives second-order expansions of both deficits under smooth deformations that preserve the $(n-1)$-th quermassintegral and converge to a ball, expressing the deficits in terms of spherical-harmonic coefficients of the boundary perturbation. A sharp, dimension-dependent lower bound $c_n=\frac{n-1}{n}$ is proved for the liminf of the deficit ratio, while an universal upper bound $\frac{\delta_T}{\delta_{AF}}\le1$ holds for all admissible sets. The results are obtained via a detailed analysis of the Monge-Ampère operator, quermassintegrals, and the spectral decomposition of boundary deformations, providing quantitative stability relationships between torsion and geometric deficits in convex geometry.

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.

Theorems & Definitions (25)

• Definition 1.1
• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.1
• Definition 2.4
• Lemma 3.1
• Theorem 3.2
• proof