THE CHEEGER CONSTANT OF A JORDAN DOMAIN WITHOUT NECKS

Gian Paolo Leonardi; Robin Neumayer; Giorgio Saracco

THE CHEEGER CONSTANT OF A JORDAN DOMAIN WITHOUT NECKS

Gian Paolo Leonardi, Robin Neumayer, Giorgio Saracco

Mathematics

Research output: Contribution to journal › Article › peer-review

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 Ω^r is equal to πr². 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.

49K20, 35J93, 49Q20

Original language	English (US)
Journal	Unknown Journal
State	Published - Apr 24 2017

Keywords

Cheeger constant
Cut locus
Focal points
Inner Cheeger formula

ASJC Scopus subject areas

General

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title = "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 πr2. 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.49K20, 35J93, 49Q20",

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T1 - THE CHEEGER CONSTANT OF A JORDAN DOMAIN WITHOUT NECKS

AU - Leonardi, Gian Paolo

AU - Neumayer, Robin

AU - Saracco, Giorgio

PY - 2017/4/24

Y1 - 2017/4/24

N2 - 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 πr2. 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.49K20, 35J93, 49Q20

AB - 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 πr2. 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.49K20, 35J93, 49Q20

KW - Cheeger constant

KW - Cut locus

KW - Focal points

KW - Inner Cheeger formula

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