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On the minimization of total mean curvature

In this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partialΩ} H\,dA \geq \sqrt{4πA(\partialΩ)}$ valid for any regular open set $Ω\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature and $A$ the area. We prove that this inequality holds true for axisymmetric domains which are convex in the direction orthogonal to the axis of symmetry. We also show that this inequality cannot be true in more general situations. However we prove that $\int_{\partialΩ} |H|\,dA \geq \sqrt{4πA(\partialΩ)}$ remains true for any axisymmetric domain.