We show that the maximal Cheeger set of a Jordan domain Ω without necks is the union of all balls of radius r=h(Ω)^−1 contained in Ω. Here, h(Ω) denotes the Cheeger constant of Ω, that is, the infimum of the ratio of perimeter over area among subsets of Ω, and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set Ωr is equal to π r^2. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.

The Cheeger constant of a Jordan domain without necks / Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 1432-0835. - 56:6(2017), pp. 1-29. [10.1007/s00526-017-1263-0]

### The Cheeger constant of a Jordan domain without necks

#### Abstract

We show that the maximal Cheeger set of a Jordan domain Ω without necks is the union of all balls of radius r=h(Ω)^−1 contained in Ω. Here, h(Ω) denotes the Cheeger constant of Ω, that is, the infimum of the ratio of perimeter over area among subsets of Ω, and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set Ωr is equal to π r^2. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.
##### Scheda breve Scheda completa Scheda completa (DC) 1-nov-2017
56
6
1
29
The Cheeger constant of a Jordan domain without necks / Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 1432-0835. - 56:6(2017), pp. 1-29. [10.1007/s00526-017-1263-0]
Leonardi, Gian Paolo; Neumayer, Robin; Saracco, Giorgio
File in questo prodotto:
File
CheegerFinal_rev1.pdf

accesso aperto

Descrizione: articolo principale
Tipologia: Post-print dell'autore (bozza post referaggio)
Dimensione 471.43 kB
##### Pubblicazioni consigliate

Caricamento pubblicazioni consigliate 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/11380/1147209`
##### Citazioni
• ND
• 15
• 12