## 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.

**Proof:** Suppose is a parametrization for , and we may assume that is parametrized with constant speed, i.e. on . Denote by the coordinate functions, i.e. . Then can be extended periodically to the whole and we have

Note that all these function are real valued. Parseval’s identity gives

By Green’s formula we have

and this is because are real valued, so and

Then we have

Therefore the inequality is proved. For the equality to hold we first need that when . Then we also have

so for . And we see right away that the equality holds only for the circle.