= Hurwitz proof of the planar isoperimetric inequality
{c}
Parametrize a positively oriented closed <plane curve> at constant <arc length> speed on $[0,2\pi]$, and let $c_n$ be its complex-position <Fourier coefficients>. <Green's theorem> and <Parseval's identity> give $A=\pi\sum_n n|c_n|^2$ and $L^2/(4\pi^2)=\sum_n n^2|c_n|^2$. Since $n\leq n^2$ for integers, the <planar isoperimetric inequality> follows. Equality leaves only the constant and positive first harmonic, a translated circle.
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