I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am also fairly certain that the same is true for the ball in three dimensions: given a fixed volume, the solid with that volume and minimal area is the ball.
However, I realised with some surprise that I do not know if this generalises to higher dimensions. Hence the question:
Consider all $n$-dimensional manifolds with doundary $M \subset \mathbb{R}^n$ with $\operatorname{vol}(M) = 1$. Is it true that $\operatorname{vol}(\partial M)$ is minimised by the ball (with the appriate radius)?
I think this should be true, but I am also aware that high-dimensional geometry can produce very unexpected results, so I am far from being certain.
(Because everything happens in $\mathbb{R}^n$, I believe there is a natural notion of volume on submanifolds; I denote this volume by $\operatorname{vol}$)