Find the surface enclosing the maximum volume per unit surface area, I congruent V/S. The solution is a sphere, which has I_sphere = (4/3 π r^3)/(4π r^2) = 1/3 r. The fact that a sphere solves the isovolume problem was only proved as recently as 1882 by Schwarz.

Dido's problem | double bubble | isoperimetric problem | sphere | surface area | volume