Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. P. Leonardi

Best constants for the isoperimetric inequality in quantitative form

created by cicalese on 30 Dec 2010
modified by leonardi on 31 Jan 2024


Published Paper

Inserted: 30 dec 2010
Last Updated: 31 jan 2024

Journal: J. Eur. Math. Soc. (JEMS)
Volume: 15
Number: 3
Pages: 1101-1129
Year: 2013

ArXiv: 1101.0169 PDF


We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992).

Keywords: isoperimetric inequality, best constants, quasiminimizers of the perimeter