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 .
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.