I want to calculate explicitly Cheeger constant for $S^2$, but I haven't found any sources or examples. I'm using this definition $$h(M)=\inf_A\{\frac{vol_{n-1}(\partial A)}{vol_n{(A)}}:vol_n(A)\leq \frac{1}{2} vol_n(M)\}$$
Here $M$ is manifold.
The only thing I found in the Internet was estimation, but not explicit answer. Is there any ways to get it?
I would appreciate any hints.
By the isoperimetric inequality on the sphere, a disk (spherical cap) has the smallest perimeter among all sets of given area on the sphere. It remains to minimize the ratio over circles of different radius. The isoperimetric inequality makes this easy by giving a convenient relation between the area and perimeter of a spherical cap: $p=\sqrt{a(4\pi -a)}$ where $a$ is the area and $p$ is the perimeter. The function $$ \frac{p}{a} = \frac{\sqrt{a(4\pi -a)}}{a} = \sqrt{\frac{4\pi -a}{a}},\quad 0<a\le 2\pi $$ is minimized by $a=2\pi$.
So, the Cheeger constant is $1$, attained by half-sphere. (Area $2\pi$, perimeter $2\pi$.)