Sharp lower bound for the Monge-Amp\`ere torsion on convex sets

TL;DR

This paper establishes a sharp lower bound for the Monge-Ampère torsion deficit in convex sets, linking it to geometric deficits and demonstrating bounds that depend only on the dimension.

Contribution

It introduces a novel lower bound for the Monge-Ampère torsion deficit using shape derivatives, connecting it to the Alexandrov-Fenchel inequality.

Findings

The ratio between the torsion deficit and the geometric deficit is bounded from below by a dimension-dependent constant.

The ratio is also bounded from above by a universal constant.

The results hold for smooth convex sets converging to a ball in ^n.

Abstract

The \emph{Monge-Amp\`ere} torsion deficit of an open, bounded convex set $\Omega\subset\R^n$ of class $C^2$ is the normalized gap between the value of the torsion functional evaluated on $\Omega$ and its value on the ball with the same $(n-1)$-quermassintegral as $\Omega$. 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.