Question: Defining on $R^3$, $V = \iiint\_S dx \, dy \, dz $ as the volume of surface $S$, with $S$ closed, bounded and arc-connected. Which is the $S$ of minimal area, that contains $V$. I know it's a bit general, so maybe you could think of another restriction without losing too much generality

Answer: 
The minimal surface [is a sphere](http://en.wikipedia.org/wiki/Isoperimetric_inequality#Isoperimetric_inequality_in_higher_dimensions).
The surface of minimum area for given volume is a Delaunay Unduloid.

The meridians have constant mean curvature with respect to the axis of rotational symmetry.

Among the unduloids, the sphere is the smallest.

It is straight forward to set up the isoperimeric problem, and then solve it by Euler-Lagrange Equations.

In this connection it is to be recalled that the locus of an ellipse focus as it rolls on a straight line without slip corresponds to this meridian. I read about it first from Blashke's book on differential geometry ( German ).