Geodesic
On the quantitative isoperimetric inequality in the plane
Bianchini, Chiara1 idref ; Croce, Gisella2 ; Henrot, Antoine3 idref
1Dipartimento di Matematica ed Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy.
2Normandie Univ, France; ULH, LMAH, FR CNRS 3335, 25 rue Philippe Lebon, 76600 Le Havre, France.
3Institut Élie Cartan de Lorraine UMR CNRS 7502, Université de Lorraine, BP 70239, 54506 Vandoeuvre-les-Nancy cedex, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 517-549

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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 of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

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DOI : 10.1051/cocv/2016002
Classification : 28A75, 49J45, 49J53, 49Q10, 49Q20
Keywords: Isoperimetric inequality, quantitative isoperimetric inequality, isoperimetric deficit, Fraenkel asymmetry, rearrangement, shape derivative, optimality conditions

Bianchini, Chiara  1   ; Croce, Gisella  2   ; Henrot, Antoine  3

1 Dipartimento di Matematica ed Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy.
2 Normandie Univ, France; ULH, LMAH, FR CNRS 3335, 25 rue Philippe Lebon, 76600 Le Havre, France.
3 Institut Élie Cartan de Lorraine UMR CNRS 7502, Université de Lorraine, BP 70239, 54506 Vandoeuvre-les-Nancy cedex, France. 
Bianchini, Chiara; Croce, Gisella; Henrot, Antoine. On the quantitative isoperimetric inequality in the plane. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 517-549. doi: 10.1051/cocv/2016002
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     title = {On the quantitative isoperimetric inequality in the plane},
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     pages = {517--549},
     year = {2017},
     publisher = {EDP-Sciences},
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