The quantitative isoperimetric inequality: old and new

I will give an account of old and new results about quantitative isoperimetry. The issue of quantifying the stability in the isoperimetric inequality has attracted the interest of mathematicians since the beginning of the last century. Even though several problems still remain unsolved, there have been some remarkable achievements in the very last years, obtained via symmetrization techniques and optimal transport.

Here I will present a new, variational method called "Selection Principle", which can be successfully applied to the analysis of quantitative forms of the isoperimetric inequality. The method is based on a penalization argument combined with the regularity theory for quasiminimizers of the perimeter. Some applications of the Selection Principle will be presented: first, a new proof of the sharp quantitative isoperimetric inequality in R^n, with an improvement on the estimate of the optimal asymptotic constant appearing in the inequality; second, the proof of a conjecture due to Hall (1992) about the precise value of the above-mentioned constant in dimension 2. Finally, I will briefly discuss other applications of the method (in particular, the quantitative stability for the double bubble) as well as some open problems.