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.

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The quermassintegral inequalities for k-convex starshaped domains

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