On The Quantitative Isoperimetric Inequality In The Plane

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

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