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 $c_{1},...,c_{m}$ in the inequality\[\\delta P(E) \geq \sum_{k=1}^{m}c_{k}\alpha(E)^{k} + o(\alpha(E)^{m}),\]valid for each Borel set $E$ with positive and finite area, with $\delta(E)$ and $\alpha(E)$ 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 $\frac{\delta P(E)}{\alpha(E)^{2}}$, we describe the general technique upon which our $2$-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in \cite{CicLeo10}.

Best constants for the isoperimetric inequality in quantitative form / Cicalese, Marco; Leonardi, Gian Paolo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 15:(2013), pp. 1101-1129. [10.4171/JEMS/387]

Best constants for the isoperimetric inequality in quantitative form

LEONARDI, Gian Paolo
2013

Abstract

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 $c_{1},...,c_{m}$ in the inequality\[\\delta P(E) \geq \sum_{k=1}^{m}c_{k}\alpha(E)^{k} + o(\alpha(E)^{m}),\]valid for each Borel set $E$ with positive and finite area, with $\delta(E)$ and $\alpha(E)$ 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 $\frac{\delta P(E)}{\alpha(E)^{2}}$, we describe the general technique upon which our $2$-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in \cite{CicLeo10}.
15
1101
1129
Best constants for the isoperimetric inequality in quantitative form / Cicalese, Marco; Leonardi, Gian Paolo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 15:(2013), pp. 1101-1129. [10.4171/JEMS/387]
Cicalese, Marco; Leonardi, Gian Paolo
File in questo prodotto:
File Dimensione Formato  
jems-2013vol15n3.pdf

non disponibili

Tipologia: Versione dell'editore (versione pubblicata)
Dimensione 232.88 kB
Formato Adobe PDF
232.88 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

Licenza Creative Commons
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/928890
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 10
  • ???jsp.display-item.citation.isi??? 9
social impact