# The isoperimetric inequality

Theorem 4.1.1 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 we have the isoperimetric inequality

$$\mathcal A\leq \dfrac{l^2}{4\pi}$$

with equality if and only if $\Gamma$ is a circle.

The heart of the Theorem 4.1.1 is parametrization, and we now declare some concepts mentioned above.

Definition 4.1.2 A parametrized curve $\gamma$ is a mapping

$$\gamma:[a,b]\to \mathbb R^2$$

then $\Gamma=\gamma([a,b])$ is a curve which is the image of $\gamma$. The curve $\Gamma$ is said to be simple if it does not intersect ifself, and closed if its two end-points coincide. That is to say

$$\gamma(s_1)\neq \gamma(s_2)$$

unless $s_1=a,s_2=b$, where $\gamma(a)=\gamma(b)$. We also impose it is regular, which means

$$\gamma\in C^1([a,b])$$

$$\gamma'\neq 0,\ \forall s\in[a,b]$$

Clearly, the conditions that $\Gamma$ be closed and simple are independent of the chosen parametrization.

Definition 4.1.3 If $\Gamma$ is parametrized by $\gamma(s)=(x(s),y (s))$, then the length $l$ of the curve $\Gamma$ is defined by

$$l=\int^b_a|\gamma'(s)|\mathrm ds=\int^b_a(x'(s)^2+y'(s)^2)^{1/2}\mathrm ds$$

The equation is guaranteed by the smoothness of $\gamma$. And the length of $\Gamma$ is a notion intrinsic to the curve, not depending on its parametrization. So we can use a parametrization by arclength $\gamma$ to parametrize $\Gamma$ if $|\gamma'(s)|=1$ for all $s$.

Definition 4.1.4 The area $\mathcal A$ of the simple closed curve $\Gamma$ is given by the formula of the calculus

$$\mathcal A=\dfrac 12\left|\int_{\Gamma}(x\mathrm dy-y\mathrm dx)\right|=\dfrac 12 \left|\int^b_ax(s)y'(s)-y(s)x'(s)\mathrm ds\right|$$

At the beginning of the proof, we give Wirtinger inequation

Lemma 4.1.5 Suppose $f\in C^1(S^1)$, then we have Wirtinger inequation

$$\int^\pi_{-\pi}(f-\bar f)^2\mathrm dx\leq \int^\pi_{-\pi}f'^2\mathrm dx$$

where

$$\bar f=\dfrac1{2\pi}\int^\pi_{-\pi}f\mathrm dx$$

with equality if and only if its Fourier coefficient

$$a_n=b_n,\ n\geq 2$$

that is to say

$$f=\bar f+a\cos x+b\sin x$$

# Weyl's equidistribution theorem

Definition 4.2.1 If $x\in \mathbb R$, we let $[x]$ denote the greatest integer less than or equal to $x$ and call the quantity $[x]$ the integer part of $x$. The fractional part of $x$ is then defined by

$$[0,1)\ni \langle x\rangle=x-[x],\ \forall x\in\mathbb R$$

The rational number case is trivial, while for irrational number case Kronecker told that if $\gamma\in\mathbb R-\backslash \mathbb Q$, then $\{\langle n\gamma \rangle \}$ is dense in the interval $[0,1)$. There is a more powerful result below.

Definition 4.2.2 A sequence of numbers $\{\xi_n\in [0,1)\}^\infty_{n=1}$ is said to be equidistributed if for every interval $(a,b)\subset [0,1)$

$$\lim_{N\to\infty}\dfrac{\#\{1\leq n\leq N:\xi_n\in(a,b)\}}{N}=b-a$$

where $\# A$ denotes the cardinality of the finite set $A$.

Theorem 4.2.3 If $\gamma\in\mathbb R\backslash\mathbb Q$, then the sequence of fractional parts $\{\langle n\gamma\rangle\}$ is equidistributed in $[0,1)$, which is Weyl's equidistribution theorem.

Lemma 4.2.4 If $f\in C(S^1(1))$, and $\gamma\in \mathbb R\backslash \mathbb Q$, then

$$\dfrac 1N\sum^N_{n=1}f(n\gamma)\to \int^1_0f(x)\mathrm dx,\text{ as } N\to\infty$$

Corollary 4.2.5 If $f\in\mathcal R(S^1(1))$, and $\gamma\in\mathbb R\backslash\mathbb Q$, then

$$\dfrac1N\sum^N_{n=1}f(n\gamma)\to \int^1_0f(x)\mathrm dx,\text{ as }N\to\infty$$