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?