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Comment by Jun8 | Hacker News Reader

This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely (http://math.stackexchange.com/a/14632/14643).

Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.

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The converse is impossible in any context where the isoperimetric inequality is applicable.

Good point. Though the Wikipedia article on it gives the most general form as requiring our region to have closure with finite Lebesgue measure. This is necessary of course by the example another corner brought up about swapping the inside and outside. But that means it exactly can't rule out the existence of regions with infinite volume (outside and inside) and finite surface area!

My intuition is that the converse is impossible in general. There is an axis along which an object with infinite volume is infinite, and taking cross sections along this axis you should be able to prove that the object has infinite area.

Section 7.1.3 of the article has a proof that such a counterexample cannot exist.

Of course, this being mathematics, changing the rules can produce a counterexample. A simple way of doing that is by declaring the inside of the shape the outside and vice versa.

I expect one could construct a counterexample around a black hole, too.

ah, thank you. i did not really understand why this was really any more "surprising" than the 2D analogue (finite area under an infinite curve). there are at least some other people who think it's not :-)

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