TITLE: Area of a elliptic surface confined by a sphere
QUESTION [5 upvotes]: Let $S$ be a surface enclosed inside the unit sphere in $R^3$. If every point of S is elliptic, then must $\operatorname{Area}(S)≤\operatorname{Area}(S^2)$?

REPLY [4 votes]: I do not think so.  Just imagine that you peel a large orange whose surface area is much bigger than that of a unit sphere. Then you "spiral" the peel to make it arbitrarily small and place it inside a unit sphere. Imagine a surface that looks more or less as this one:

It has positive curvature so very point is elliptic. 
Here you remove a small cylinder around the vertical axis so there is no problems with the curvature.
Since you can have as many "turns" as you want, its surface area can be arbitrarily large while the surface occupies a small region in space.