Geodesic
Best constants for the isoperimetric inequality in quantitative form
Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 1101-1129 Cet article a éte moissonné depuis la source EMS Press

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We prove some results in the context of isoperimetric inequalities with quantitative terms. In the 2-dimensional case, our main contribution is a method for determining the optimal coefficients c1​,...,cm​ in the inequality δP(E)≥∑k=1m​ck​α(E)k+o(α(E)m), valid for each Borel set E with positive and finite area, with δP(E) and α(E) being, respectively, the \textit{isoperimetric deficit} and the \textit{Fraenkel asymmetry} of E. In n dimensions, besides proving existence and regularity properties of minimizers for a wide class of \textit{quantitative isoperimetric quotients} including the lower semicontinuous extension of α(E)2δP(E)​, we describe the general technique upon which our 2-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in [12].
DOI : 10.4171/jems/387
Classification : 52-XX, 28-XX, 49-XX, 00-XX
Keywords: Best constants, isoperimetric inequality, quasiminimizers of the perimeter 
@article{JEMS_2013_15_3_a14,
     author = {Marco Cicalese and Gian Paolo Leonardi},
     title = {Best constants for the isoperimetric inequality in quantitative form},
     journal = {Journal of the European Mathematical Society},
     pages = {1101--1129},
     year = {2013},
     volume = {15},
     number = {3},
     doi = {10.4171/jems/387},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/387/}
}
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Marco Cicalese; Gian Paolo Leonardi. Best constants for the isoperimetric inequality in quantitative form. Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 1101-1129. doi: 10.4171/jems/387

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