  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).
