## 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.

For a compact set we denote

We say that the set has Minkowski content if the limit

exists. It can be proved that if is a quasi-simple curve then the Minkowski content of exists if and only if is rectifiable, and in this case , where is the length of the curve.

Suppose now that is open, bounded and its boundary is a simple rectifiable curve . Then we have .

**Proof:** Denote

and

We see right away that and ., and this union is disjoint. We can see that

where the is the Minkowski sum of the corresponding sets and is the unit open disk of radius . Now we apply the Brunn-Minkowski inequality and get

Adding these two inequalities we get that

Divide this inequality by and take the as to obtain

which is just the isoperimetric inequality.

I wonder if this proof is the basically the same as the one Levy has given in 1922:

https://archive.org/stream/leconsdanalysefo00levyrich#page/n281/mode/2up

I’m not sure. The proof in the post is based on Brunn Minkowski. The proof in the book you suggest does not contain many details. The idea of moving a segment parallel in the interior of the domain is related to Minkowski content.

I found the reference to Levy in this book by Ledoux and Talagrand, who state that his proof has not been understood for a long time, to which I agree: https://books.google.de/books?id=fuclBQAAQBAJ&pg=PA34&dq=%22has+not+been+understood+for+a+long+time%22&hl=de&sa=X&ved=0ahUKEwiNkeDUs6TSAhXEPhQKHe-1BE4Q6AEIJjAB#v=onepage&q=%22has%20not%20been%20understood%20for%20a%20long%20time%22&f=false