Show that the isoperimetric inequality is equivalent to Wirtinger's inequality, which says that if $f$ is $2\pi$-periodic, of class $C^1$, and satisfies $\int_0^{2\pi}f(t)dt=0$, then $$\int_0^{2\pi}|f(t)|^2dt\leq \int_0^{2\pi}|f'(t)|^2dt$$ with equality if and only if $f(t) = A\sin t + B \cos t$

Part of the proof:
(Isoperimetric $\implies$ Wirtinger)
If we re-parametrize $t = ks$ with $k = T/(2\pi)$, then the change of variable shows that it is sufficient to prove the claim for the period being $2\pi$.
Given f with mean 0. we found $F(t) :=\int_0^t f(s) ds$ is a $2\pi$-periodic function. So the isoperimetric and Hölder inequalities imply $$\int_0^{2\pi}f^2(t)dt=\int_0^{2\pi}f(t)F'(t)dt$$ $$ \leq \frac 1 {4\pi}\left (\int_0^{2\pi}\sqrt {(f'(t))^2 + (F'(t))^2}dt\right )^2 \tag{1}$$ $$ \leq \frac 1 2 \int_0^{2\pi}(f'(t))^2 +f(t)^2 dt \tag{2}$$ which is equivalent to the Wirtinger's inequality (How?).
(Is $\frac 1 2 \int_0^{2\pi}(f'(t))^2 +f(t)^2 dt$ equivalent to $\int_0^{2\pi}|f'(t)|^2dt$? )

Could you also show how to arrive $(1)$ and $(2)$ ?

Some defs and thms:

  • If $\Gamma$ is parametrized by $\gamma (s) = (x(s), y(s))$, then the length of the curve $\Gamma$ is defined by $l=\int_a^b |\gamma '(s)|ds=\int_a^b{x'(s)^2+ y'(s)^2}ds$

  • (the isoperimetric inequality): Suppose that $\Gamma$ is a simple closed curve in $\mathbb R^2$ of length $l$, and let $\mathcal A$ denote the area of the region enclosed by this curve. Then $$\mathcal A\leq \frac {l^2}{4\pi}$$ with equality if and only if $\Gamma$ is a circle.

  • the area $\mathcal A $ of the region enclosed by a simple closed curve $\Gamma$ is given by $\frac12 \left |\int_a^b (x(s)y'(s)-y(s)x'(s))\,ds\right |$


For step 1, use isoperimetric inequality: $$\mathcal A \le \frac{l^{2}}{4\pi}$$ Which can also be written as $$\int _{a}^{b}x(s)y^{'}(s)ds \le \int _{a}^{b}(x^{'}(s)^{2}+y^{'}(s)^{2})^{1/2}ds.$$ if $\Gamma$ is parametrized by $\gamma=(x(s),y(s))$ Then substitute $f(t)=x(t)$ and $F^{'}(t)=y^{'}(t)$. For step 2, use Holder inequality: $$\int_{a}^{b}| f(x)g(x) |dx \le (\int_{a}^{b} | f(x)|^{p}dx)^{1/p}(\int_{a}^{b}| g(x)| ^{q}dx)^{1/q}$$ For $(q, p) \in (1,\infty), \frac{1}{p}+\frac{1}{q}=1$

Substitute $g(t)=1$, $f(t)=\sqrt{f^{'}(t)^{2}+F^{'}(t)^{2}}dt$ and $p=q=2$ to get

$$\int_{0}^{2\pi}\sqrt{f^{'}(t)^{2}+F^{'}(t)^{2}} \le \sqrt{2\pi} \sqrt{\int_{0}^{2\pi}(f^{'}(t))^{2}+(f(t))^{2}dt.} $$


$$\int_{0}^{2\pi} f^{2}(x)dt\le \frac{1}{2}\int_{0}^{2\pi} (f^{'}(t))^{2}+(f(t))^{2}dt \to \frac{1}{2}\int_{0}^{2\pi}f^{2}(t)dt\le \frac{1}{2}\int_{0}^{2\pi} (f^{'}(t))^{2}dt.$$ Which is just Wirtinger's inequality.


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