## Proof of the Isoperimetric Inequality 5

Suppose is a simple, sufficiently smooth closed curve in . If is its length and is the area of the region it encloses then the following inequality holds

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 is simple if is injective. It is a closed simple curve if and is injective on . We say that a curve is quasi-simple if it the mapping is injective with perhaps finitely many exceptions.

## The Brunn-Minkowski Inequality

The Brunn-Minkowski Inequality gives an estimate for the measure of the set in terms of the Lebesgue measures of and in (of course, we can only speak of such an inequality if all the sets are Lebesgue measurable). The inequality has the form

## Existence of Maximal Area Polygon with given sides

Suppose are given positive real numbers such that there exists a polygon with vertices having 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 has length and encloses area . Then

and equality holds if and only if is a circle.

## Cyclic polygon has largest area

Suppose are positive real numbers. Then the following statements are equivalent:

1. satisfy the inequalities

2. There exists a polygon with sides .

3. There exists a convex polygon with sides .

4. There exists a cyclic polygon with sides .

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

## Isoperimetric inequality for polygons

Among polygons with sides with given perimeter , the one which maximizes the area is regular.