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

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