Is there a shape with infinite volume but finite surface area?

Question

Is there any pathological shape that has a finite surface area but an infinite volume, sort of like the opposite of a Gabriel's horn?

Answer 1

Nope!

See the "Converse" section of http://en.wikipedia.org/wiki/Gabriel's_Horn .

Answer 2

Aside from my comment about all of $\Bbb R^3$, there is none. A sphere has the smallest surface for a given volume. One pathological case would be fractals that have well defined volume but no well defined surface area. In most cases you would like to say they have infinite area.

Answer 3

Imagine a sphere outline in an infinite void. If the area within the sphere outline is empty space, and the space outside is solid, it is a 3D shape of infinite volume, and since it continues infinitely, there is no outer edge of the shape to apply surface area to, meaning the surface area is a finite value, on the same spherical plane as the outline.