In 1970, Cheeger defined the well-known Cheeger constant for general manifold and proved a lower bound estimation for the eigenvalue. Actually I want to compute the Cheeger constant for some special domain to get a further understanding about this definition.
Recall the Cheeger constant for planar domain is defined by $$ C(U):=\inf_{A\subset U, V(A)\leq\frac{1}{2}V(U)}\frac{P(\partial A-\partial U)}{V(A)}, $$ where $V(\cdot)$ denotes the 2-dim volume and $P(\cdot)$ denotes the 1-dim measure.
Now I want to compute the Cheeger constant for unit disc $D\subset\mathbb{R}^2$. I already proved that, if $A$ is such a minimizer, then $P(\partial A-\partial D)\leq 2$, and $\partial A-\partial D$ must touch $\partial D$ on only two point. Intuitively, such $A$ should be half of the disc and $\partial A-\partial D$ should be the diameter.
Here is my attempt. If we fix two points in $\partial D$ and consider a curve $\gamma$ between these two points with given length $l$, then we know if the area of $A$ is biggest, then $A$ must be a part of circle, where $A$ is the domain enclosed by this curve and $\partial D$. I want to show that in this case, the quotient given in the above definition is bigger than the case that $\gamma$ is some line segment. But the computation is too tedious to go further.
I wonder if I miss something important, or there is another method to solve it. Any help will be appreciated a lot! Thanks!