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The property is true for $n =3$. But is it also true for $n \gt 3$?

Is the object in $R^n$ with minimal surface area for a given volume, the higher dimensional analog of a sphere?

Lelouch
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    This is the isoperimetric inequality: see https://en.wikipedia.org/wiki/Isoperimetric_inequality#Isoperimetric_inequality_in_.7F.27.22.60UNIQ--postMath-00000019-QINU.60.22.27.7F – Mariano Suárez-Álvarez Mar 04 '17 at 17:15
  • So, for equality, the subset S must be a ball in $R^n$, which is the higher homologue of a 3d sphere. Is that correct? – Lelouch Mar 04 '17 at 17:20

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