Surface areas of different submanifolds of $\Bbb R^3$

Question

Can you circumscribe a continuous, smooth manifold in $\Bbb R^3$ with another manifold that completely encapsulates it but has a surface area which smaller than that of the one contained? Is there a theorem that says yes or no?

Answer 1

Let's first go down a dimension, and circumscribe a continuous, smooth, closed curve in $\mathbb{R}^2$ with another closed curve that completely encapsulates it, but has length smaller than the one it contains. Namely, take a high iteration of the approximations to the Peano curve, smooth off their corners, and close them up. One obtains continuous, smooth, closed curves all in the same compact region, with lengths approaching infinity. But that compact region can be circumscribed by a single circle.

Perhaps you can now convince yourself that this construction may be bumped up a dimension, to give a counterexample that answers your question in the negative.