I'm trying to find a reference about or related to the Cheeger constant of a ball in $\mathbb{R}^3$. The definition of Cheeger constant $h(\Omega)$ is as follows: given a domain $\Omega\subset\mathbb{R}^d$,
$$h(\Omega) = \inf \{\frac{|\partial D|}{|D|}, D\subset \Omega\}$$ where we use Lebesgue measures respectively of dimension $d-1$ for $\partial D$ and dimension $d$ for $D$. And $D$ is varying among the smooth subdomains of $\Omega$. Is there any reference on how to compute this for balls of some size $r>0$ ($\{|x|\le r\}$) in $\mathbb{R}^3$.
This is related to a problem of first non-linear eigenvalue of $\textbf{1-Laplacian}$ on balls in $\mathbb{R}^3$ that I'm working on. Any suggestion will be appreciated!
The isoperimetric inequality tells us that the optimizer is a ball of the largest possible radius, so the Cheeger constant is $$\frac{4\pi r^2}{\frac43 \pi r^3}= \frac3r$$