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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 $\\pi r^2$. 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