Geodesic
The Quantitative Isoperimetric Inequality for Planar Convex Domains
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 573-589 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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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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     author = {Nitsch, Carlo},
     title = {The {Quantitative} {Isoperimetric} {Inequality} for {Planar} {Convex} {Domains}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {573--589},
     year = {2008},
     volume = {Ser. 9, 1},
     number = {3},
     zbl = {1190.26025},
     mrnumber = {2455332},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a2/}
}
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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/