Consider a cylinder with a dome. What dimensions maximize volume with respect to total surface area? Is it the largest vaguely stable shape? Vaguely stable meaning flat bottom, sides that rise perpendicularly from the bottom, and convexly curving in, like most bottles. To add one additional parameter, make the dome an ellipsoid.
A related problem: Given a circular field, construct a dome over it that maximizes the volume inside with respect to the surface area of the dome. Is it a section of a sphere (spherical cap)? Is it a hemisphere?
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