Best constants for the isoperimetric inequality in quantitative form

Abstract

We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992).