THE CHEEGER CONSTANT OF A JORDAN DOMAIN WITHOUT NECKS

Gian Paolo Leonardi, Robin Neumayer, Giorgio Saracco

Research output: Contribution to journalArticlepeer-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 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

Original languageEnglish (US)
JournalUnknown Journal
StatePublished - Apr 24 2017

Keywords

  • Cheeger constant
  • Cut locus
  • Focal points
  • Inner Cheeger formula

ASJC Scopus subject areas

  • General

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