Geodesic
The Quantitative Isoperimetric Inequality for Planar Convex Domains
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 573-589.

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We prove that among all the convex bounded domains in $\mathbb{R}^2$ having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domains. 
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Nitsch, Carlo. The Quantitative Isoperimetric Inequality for Planar Convex Domains. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 573-589. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a2/

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