### Archive

Posts Tagged ‘isoperimetric inequality’

## Proof of the Isoperimetric Inequality 5

Suppose ${\Gamma}$ is a simple, sufficiently smooth closed curve in ${\Bbb{R}^2}$. If ${L}$ is its length and ${A}$ is the area of the region it encloses then the following inequality holds

$\displaystyle 4 \pi A \leq L^2.$

I am going to present another proof of the isoperimetric inequality. This time with Fourier series.

## Minkowski content and the Isoperimetric Inequality

The proof of the isoperimetric inequality given below relies on the Brunn-Minkowski inequality and on the concept of Minkowski content.

In what follows we say that a curve parametrized by ${z(t)=(x(t),y(t)),\ t \in [a,b]}$ is simple if ${t\mapsto z(t)}$ is injective. It is a closed simple curve if ${z(a)=z(b)}$ and ${z}$ is injective on ${[a,b)}$. We say that a curve is quasi-simple if it the mapping ${z}$ is injective with perhaps finitely many exceptions.

## The Brunn-Minkowski Inequality

May 9, 2012 1 comment

The Brunn-Minkowski Inequality gives an estimate for the measure of the set ${A+B=\{a+b : a \in A,b \in B\}}$ in terms of the Lebesgue measures of ${A}$ and ${B}$ in ${\Bbb{R}^d}$ (of course, we can only speak of such an inequality if all the sets ${A,B,A+B}$ are Lebesgue measurable). The inequality has the form

$\displaystyle |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}.$

## Existence of Maximal Area Polygon with given sides

Suppose ${a_1,..a_n,\ n \geq 3}$ are given positive real numbers such that there exists a polygon with ${n}$ vertices having ${a_1,..,a_n}$ as lengths of its size. Then among all polygons having these given side lengths there exists one which has maximal area.

Since I have used in the linked post the fact that the maximal area polygon with given sides exists, I will give a proof in the following.

## Proof of the Isoperimetric Inequality 3

I will present here a third proof for the planar Isoperimetric Inequality, using some simple notions of differential curves. For this suppose that the simple closed plane curve ${C}$ has length ${L}$ and encloses area ${A}$. Then

$\displaystyle L^2 \geq 4 \pi A$

and equality holds if and only if ${C}$ is a circle.

## Cyclic polygon has largest area

Suppose ${a_1,..,a_n}$ are positive real numbers. Then the following statements are equivalent:

1. ${a_1,...,a_n}$ satisfy the inequalities

$\displaystyle 2a_i < a_1+...+a_n,\ i=1..n$

2. There exists a polygon with sides ${a_1,..,a_n}$.

3. There exists a convex polygon with sides ${a_1,..,a_n}$.

4. There exists a cyclic polygon with sides ${a_1,..,a_n}$.

Of all these, the cyclic polygon has the maximal area.

Categories: shape optimization

## Isoperimetric inequality for polygons

Among polygons with ${n \geq 3}$ sides with given perimeter ${P}$, the one which maximizes the area is regular.