*Submitted Paper*

**Inserted:** 19 feb 2015

**Last Updated:** 19 feb 2015

**Year:** 2014

**Links:**
http://arxiv.org/abs/1406.6984

**Abstract:**

In this paper we are interested in possible extensions of an inequality due to Minkowski:
$\int_{\partial\Omega} H\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ from convex smooth sets to any regular open set
$\Omega\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature of $\partial\Omega$ 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_{\partial\Omega}

H

\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ remains true for any axisymmetric
domain.