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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?

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Sean: Out of curiosity, are you just reading up on isoperimetric inequalities, or planning to use this to prove new ones? – Anton Lukyanenko Jan 27 '12 at 3:58
Reading up, mainly, and learning about Fourier analysis and representation theory. – Sean Eberhard Jan 27 '12 at 8:37
up vote 13 down vote accepted

For the second time today (see Bodies of constant width?), 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.

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Awesome. Thanks. – Sean Eberhard Jan 26 '12 at 22:28

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