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The Koch Snowflake has infinite perimeter and finite area. Gabriel's Horn has infinite surface area and finite volume. I'm interested in existence of examples illustrating the converse of the behavior of the snowflake and the horn, except for $n$ dimensions.

For the $n=2$ case (infinite area and finite perimeter), the isoperimetric inequality gives us a nice proof refuting the existence of such a shape, as shown in this question.

Apparently the $n=3$ case (infinite volume and finite surface area) does not possess such an example either, and it's discussed without proof here.

(Note: I'm not interested in degenerate cases like $\mathbb{R}^3$ minus a ball, but I'm not sure how to formally exclude such cases.)

Main question: Is there a way to prove a statement about nonexistence for all $n\geq2$ ?

AegisCruiser
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  • I think the same issue arises as with the linked $n=2$ question. If you insist your region is bounded to get rid of degenerate cases, then it obviously will have finite volume. – Cheerful Parsnip Apr 20 '17 at 02:40
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    My guess is it depends on the interpretation of the word "shape". For example, the exterior of a circle in $2D$ has infinite $2$-volume but finite $1$-volume (circumference)? – Vlad Apr 20 '17 at 03:30
  • You're right, insisting on boundedness of the region is far too strong of a condition--it eliminates all the interesting cases (including Gabriel's Horn). So at it's heart, my original question does indeed require a more rigorous definition of "shape."

    I also see two classes of the original examples: the snowflake is bounded but not smooth--this will not help in searching for a converse example. The horn is unbounded and smooth--we want to look in this direction.

    Perhaps we allow unboundedness, but only in one direction? (Whatever that means...)

     – AegisCruiser Apr 20 '17 at 05:43

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