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I learned that the sphere has the smallest total mean curvature among all convex solids with a given surface area. This actually implies the sphere also has the smallest total mean curvature among all convex solids with a given volume. The first result can be proved by using Steiner symmetrization, and the latter result is a consequence of of the first one and the isoperimetric inequality.

Now my question is, does sphere have the smallest total mean curvature among all mean convex solids with a given volume?

By mean convex set, I mean bounded smooth set with nonnegative mean curvature on the boundary. I searched but didn't find any satisfying result. It seems that people haven't considered minimizing total mean curvature under the mean convex set setting. Also, the condition of fixing the volume should be added, since I believe if only fixing the surface area then the infimum of total mean curvature may not be attained. Any ideas on how to stab on this question? Is it a trivial question?

Any comment and ideas would be really appreciated.

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Even with an area constraint, the minimization of total mean curvature for mean-convex surfaces seems to be open. You should look at this recent article http://link.springer.com/article/10.1007/s12220-015-9646-y

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