Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - R. Neumayer - G. Saracco

The Cheeger constant of a Jordan domain without necks

created by neumayer on 24 Apr 2017
modified by saracco on 01 Mar 2018


Published Paper

Inserted: 24 apr 2017
Last Updated: 1 mar 2018

Journal: Calc. Var. Partial Differential Equations
Volume: 56
Number: 6
Pages: 164
Year: 2017
Doi: 10.1007/s00526-017-1263-0

ArXiv: 1704.07253 PDF


We show that the maximal Cheeger set of a Jordan domain $\Omega$ without necks is the union of all balls of radius $r = h(\Omega)^{-1}$ contained in $\Omega$. Here, $h(\Omega)$ denotes the Cheeger constant of $\Omega$, that is, the infimum of the ratio of perimeter over area among subsets of $\Omega$, 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 $\Omega^r$ is equal to $\pi r^2$. 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.

Keywords: cut locus, Cheeger constant, focal points, inner Cheeger formula