On the Minimization of Total Mean Curvature

In this paper we are interested in possible extensions of an inequality due to Minkowski: \(\int _{\partial \Omega } H\,dA \ge \sqrt{4\pi A(\partial \Omega )}\) 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 \(\int _{\partial \Omega } |H|\,dA \ge \sqrt{4\pi A(\partial \Omega )}\) remains true for any axisymmetric domain.