  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.
