Mathematics

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--M\"uller 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).