# The Cheeger constant of a Jordan domain without necks

## TL;DR

This paper characterizes the maximal Cheeger set of a Jordan domain without necks as the union of all inscribed balls of a specific radius, linking geometric properties to the Cheeger constant and providing approximation methods.

## Contribution

It establishes a precise geometric description of Cheeger sets in Jordan domains without necks and connects the Cheeger constant to the area of inner parallel sets, with applications to complex shapes.

## Key findings

- Maximal Cheeger set is union of inscribed balls of radius 1/h(Ω).
- Radius r is uniquely determined by the area of the inner parallel set.
- Application to approximating the Cheeger constant of the Koch snowflake.

## Abstract

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.