In the Poincare disc model, a version of the isoperimetric inequality states that $L(\partial(A))>c\mu(A),$ where $\mu$ is the hyperbolic area and $L(\gamma)=\int_0^1 \frac{|\gamma'(t)|}{1-|\gamma(t)|^2},$ for a curve $\{\gamma(t)\}_{t=0}^1$ and for some positive constant $c$.
Is there an elementary proof of this result, and what is the best constant possible here? I guess for a fixed area the set which minimizes the size of the boundary is the disc of that given area, we can check that the constant here would be $1$.
Furthermore, if we restrict our class of sets $A$ to large sets (say, those sets $A$ which contain a ball $B(z, r)$ for some $z\in A$ and large $r$), can we make the constant c in the inequality as large as we want by taking $r$ large?
Any help or references would be greatly appreciated!
I am wondering what the Poincare disk model has to do with it, I guess you just mean the curvature = -1. (because the curvature is -1 in the Poincare disk model)
Then the circumference of a circle of radius $r$ is $ 2\pi \sinh (r)$
And the area is: $ 2\pi ( \cosh (r) - 1) $
So your formula becomes: $ \frac{\cosh (r) - 1}{r \sinh (r)} $
Plotting this function it looks to have a limit at r=0 of .5 and it is decreasing away from it all the way to 0
Hopes this helps