On the Minimization of Total Mean Curvature

In this paper we are interested in possible extensions of an inequality due to Minkowski: integral(partial derivative Omega) H dA >= root 4 pi A(partial derivative Omega) from convex smooth sets to any regular open set Omega subset of R-3, where H denotes the scalar mean curvature of partial derivative 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 integral(partial derivative Omega) vertical bar H vertical bar dA >= root 4 pi A(partial derivative Omega) remains true for any axisymmetric domain.