  In this paper we study the quantitative isoperimetric inequality in the
plane. We prove the existence of a set $\Omega$, different from a ball, which
minimizes the ratio $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the
isoperimetric deficit and $\lambda$ the Fraenkel asymmetry, giving a new proof
ofthe quantitative isoperimetric inequality. Some new properties of the optimal
set are also shown.
