Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case?


Specifically, I imagine such a proof would replace $S^1 \cong \text{SO}(2)$ with the group $\text{SO}(n+1)$, and we would decompose a smooth function on $S^n$ according to the representation theory of $\text{SO}(n+1)$, and then express the volume and surface area in terms of the "Fourier coefficients".


If this doesn't work, why not?



General Isoperimetric Inequality via Representation Theory of SO(n)
For the second time today (see [Bodies of constant width?](https://mathoverflow.net/questions/86742)), I give the answer: see "Geometric Applications of Fourier Series and Spherical Harmonics" by Helmut Groemer. That book deals exactly with your question, and the answer is yes.


