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#### L. Mazzieri

#### Minkowski inequality for mean convex domains

In response to a question raised by Huisken, we prove that the Minkowski Inequality
$$\textstyle
\big|\partial \Omega\big|^\frac{n-2}{n-1} \, | {\mathbb{S}^{n-1}}|^\frac{1}{n-1} \,\, \leq \,\, \int\limits_{\partial \Omega} \!\frac{\rm H}{n-1} \,\, {\rm d}\sigma \,
$$
holds true under the mere assumption that $\Omega$ is a bounded domain with smooth mean convex boundary sitting inside $\mathbb{R}^n$, $n \geq 3$. The result is new even for surfaces in Euclidean three-space, and can be used in this setting to deduce the celebrated De Lellis-Müller nearly umbilical estimates, with a better constant. Our proof relies on a careful analysis of the level set flow of the $p$-capacitary potentials of $\Omega$, as $p \to 1$. (Joint works with V. Agostiniani, M. Fogagnolo and A. Pinamonti).

In response to a question raised by Huisken, we prove that the Minkowski Inequality
$$
\big|\partial \Omega\big|^{(n-2)/(n-1)} \, | {\mathbb{S}^{n-1}}|^{1/(n-1)} \,\, \leq \,\, \int\limits_{\partial \Omega} \!\frac{\rm H}{n-1} \,\, {\rm d}\sigma \,
$$
holds true under the mere assumption that $\Omega$ is a bounded domain with smooth mean convex boundary sitting inside $\mathbb{R}^n$, $n \geq 3$. The result is new even for surfaces in Euclidean three-space, and can be used in this setting to deduce the celebrated De Lellis-Müller nearly umbilical estimates, with a better constant. Our proof relies on a careful analysis of the level set flow of the $p$-capacitary potentials of $\Omega$, as $p \to 1$. (Joint works with V. Agostiniani, M. Fogagnolo and A. Pinamonti).

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