Minkowski inequality for mean convex domains

Problem's Description

The classical Minkowski inequality states that, $\forall \Omega\subset \mathbb R^{n+1}$ convex with $C^2$ boundary, \begin{align} \int_{\partial \Omega} H\ge C_n |\partial \Omega|^{\frac{n-1}{n}}, \label{MI} \end{align} where $H$ is the mean curvature, $c_n$ is a dimensional constant. Equality holds if and only if $\Omega$ is a round ball.

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\noindent{\bf Open Problem}: Extend inequality (\ref{MI}) for general mean-convex domains.

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(\ref{MI}) is verified for starshaped mean-convex domains by Guan-Li and for outward area minimizing domains by Huisken.

1. Article The quermassintegral inequalities for k-convex starshaped domains

   • Pengfei Guan and Junfang Li

   Advances in Mathematics 221, 1725-1732, 2009