In response to a question raised by Huisken, we prove that the Minkowski Inequality $$\big\partial \Omega\big^{(n2)/(n1)} \,  {\mathbb{S}^{n1}}^{1/(n1)} \,\, \leq \,\, \int\limits_{\partial \Omega} \!\frac{\rm H}{n1} \,\, {\rm d}\sigma \, $$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 threespace, and can be used in this setting to deduce the celebrated De LellisM\"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).
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Minkowski inequality for mean convex domains
Research Group:
Lorenzo Mazzieri
Institution:
Università di Trento
Location:
A133
Schedule:
Thursday, January 10, 2019  14:00
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