## 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.

## Proof of the isoperimetric inequality 2

I will continue the series of proofs for the isoperimetric inequality in the two dimensional case, i.e. if a simple closed curve (which we suppose for simplicity that it consists of piecewise curves) of length encloses an area then and the equality is attained only if is the boundary of a circle.

## Proof of the Isoperimetric Inequality

The Isoperimetric inequality gives a bound for the area in terms of the perimeter of a set. It says that the greatest area that can be enclosed by a curve which has length is maximal when the curve is the boundary of a circle, or equivalently the minimum of the perimeter of a curve which encloses a set of area is attained again for the circle. In two dimension the inequality says that where is the perimeter and is the area (Lebesgue measure) of a plane region .

## Region which can sustain the Largest Sandpile

Among all plane regions which are open and simply connected and without holes of given area the circle can support the largest sandpile.

## Existence Result for the Isoperimetric Problems

The tricky part is how to define the perimeter of a Lebesgue measurable set with finite perimeter. This can be done considering the space of **bounded variation functions**, denoted . By definition we have for an open set that . Here we denoted by the space of continuously differentiable functions with compact support in . Because of the density of the space of infinitely differentiable functions with compact support in in the space , we could have replaced by in the above definition. You could take a look at this blog post for a detailed description of or at the Wikipedia page.

We say that a set of finite Lebesgue measure is a set of **finite perimeter** in if its characteristic function belongs to . This means that the distributional gradient is a vector valued measure with finite total variation. The total variation is called the **perimeter** of .

In the same way we can define the perimeter of a Lebesgue measurable set relative to an open set . We say that is a set of **finite perimeter relative to ** if the characteristic function belongs to the space .