**abstract:**
One of the Alexandrov-Fenchel inequalities states that, among convex domains $K\subseteq\mathbb R^{n$} with perimeter equal to the perimeter of the ball, the quantity $\int_{{\partial} K} H$ is minimized by the ball ($H$ denotes the mean curvature of the boundary).
This is a higher-order isoperimetric inequality: instead of comparing volume and perimeter, we are comparing perimeter and integral of the mean curvature.
The validity of the inequality is open for mean-convex (i.e., $H\ge 0$) domains.

We will consider the inequality without the assumption of mean-convexity, but replacing $H$ with its absolute value $

H

$ and restricting our study to domains $K$ which are $C^{1$}-perturbations of the ball.
Under these assumptions, we will explain why the desired inequality is "morally" equivalent to the following functional inequality: given $u:\mathbb S^{{n}-1}\to\R$ with $

u_{{C}^{1}$} sufficiently small, it holds $\int (\Delta u)

\nabla u^{2} \le \frac{n-2}{n-1}(\sup\Delta u)\int

\nabla u^{2$.
}
Thanks to this insight, one can prove the inequality in some special cases.

Wed 13 Oct, 16:30 - 18:00, Aula Dini

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