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In Infinite Volume but Finite Surface Area a question is asked whether there is some shape in $\Bbb R^3$ that have an infinite volume, but a finite surface area, sort of the opposite of Gabriel's Horn.

The answer seems to be that there is none, if there is a surface area. (which seems reasonable). Would that still hold if we extend to $\Bbb R^n$ (volume defined as $n$ dimensions, area $n-1$ dimensions)? Any other mathematical room where it would be possible?

Not sure what I should tag this question with

Lennart - Slava Ukraini
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  • My inclination is to say that it's still not possible, at least not for finite $n$. (I'd guess that it's also impossible even if $n$ is infinite, but I'm less sure about what that would mean.) – Brian Tung Jul 21 '19 at 06:16
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    As commented on the linked question, as stated all of $\Bbb R^3$ is a counterexample to the assertion (no surface area at all), as is the exterior of a ball in $\Bbb R^3$ (still infinite volume and finite surface area). The same counterexamples work in all (finite) dimensions. If one wants to rule such examples out, one needs to phrase the question with more details. – Greg Martin Jul 21 '19 at 07:05

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