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# 数学代写|实分析代写Real Analysis代考|MATH721 The isoperimetric inequality

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## 数学代写|实分析代写Real Analysis代考|The isoperimetric inequality

Let $\Gamma$ denote a closed curve in the plane which does not intersect itself. Also, let $\ell$ denote the length of $\Gamma$, and $\mathcal{A}$ the area of the bounded region in $\mathbb{R}^2$ enclosed by $\Gamma$. The problem now is to determine for a given $\ell$ the curve $\Gamma$ which maximizes $\mathcal{A}$ (if any such curve exists).

A little experimentation and reflection suggests that the solution should be a circle. This conclusion can be reached by the following heuristic considerations. The curve can be thought of as a closed piece of string lying flat on a table. If the region enclosed by the string is not convex (for example), one can deform part of the string and increase the area enclosed by it. Also, playing with some simple examples, one can convince oneself that the “flatter” the curve is in some portion, the less efficient it is in enclosing area. Therefore we want to maximize the “roundness” of the curve at each point.

Although the circle is the correct guess, making the above ideas precise is a difficult matter.

The key idea in the solution we give to the isoperimetric problem consists of an application of Parseval’s identity for Fourier series. However, before we can attempt a solution to this problem, we must define the notion of a simple closed curve, its length, and what we mean by the area of the region enclosed by it.

## 数学代写|实分析代写Real Analysis代考|Curves, length and area

A parametrized curve $\gamma$ is a mapping
$$\gamma:[a, b] \rightarrow \mathbb{R}^2$$
The image of $\gamma$ is a set of points in the plane which we call a curve and denote by $\Gamma$. The curve $\Gamma$ is simple if it does not intersect itself, and closed if its two end-points coincide. In terms of the parametrization above, these two conditions translate into $\gamma\left(s_1\right) \neq \gamma\left(s_2\right)$ unless $s_1=a$ and $s_2=b$, in which case $\gamma(a)=\gamma(b)$. We may extend $\gamma$ to a periodic function on $\mathbb{R}$ of period $b-a$, and think of $\gamma$ as a function on the circle. We also always impose some smoothness on our curves by assuming that $\gamma$ is of class $C^1$, and that its derivative $\gamma^{\prime}$ satisfies $\gamma^{\prime}(s) \neq 0$. Altogether, these conditions guarantee that $\Gamma$ has a well-defined tangent at each point, which varies continuously as the point on the curve varies. Moreover, the parametrization $\gamma$ induces an orientation on $\Gamma$ as the parameter $s$ travels from $a$ to $b$.

Any $C^1$ bijective mapping $s:[c, d] \rightarrow[a, b]$ gives rise to another parametrization of $\Gamma$ by the formula
$$\eta(t)=\gamma(s(t))$$
Clearly, the conditions that $\Gamma$ be closed and simple are independent of the chosen parametrization. Also, we say that the two parametrizations $\gamma$ and $\eta$ are equivalent if $s^{\prime}(t)>0$ for all $t$; this means that $\eta$ and $\gamma$ induce the same orientation on the curve $\Gamma$. If, however, $s^{\prime}(t)<0$, then $\eta$ reverses the orientation.

## 数学代写|实分析代写Real Analysis代考|Curves, length and area

$$\gamma:[a, b] \rightarrow \mathbb{R}^2$$

$$\eta(t)=\gamma(s(t))$$

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