Context:
I recently encountered the notion of the Cheeger constant in graph theory and in Riemannian geometry when planning for some intensive studies on expander graphs for the summer. To gain some intuition and hands-on feeling for the definition and its geometric origin, I tried computing the Cheeger constant for the unit disc in $\mathbb{R}^2$ (equipped with the induced metric), but…I can't really finish.
Question:
Simply put, I want to compute the Cheeger constant of the unit disc, the constant being defined as $$\inf_E\frac{S(E)}{\min(V(A),V(B))},$$ where $E$ is a $1$-dimensional submanifold dividing the disc into disjoint subsets $A$ and $B$, $S(E)$ denotes the $1$-dimensional volume (i.e. length) of $E$, and $V$ denotes $2$-dimensional volume (area). See the following entry if you want further detail: https://en.wikipedia.org/wiki/Cheeger_constant
Attempt:
Intuitively, I feel that the minimizing submanifold $E$ would be any diameter of the disc, in which case the constant should be $\frac{2}{\pi/2}=\frac{4}{\pi}$. I suspect this since I want to maximize the denominator, which happens when the disc is split exactly in half, and since I also want to minimize the length of $E$ (so a straight line, no curvature, is expected).
Using ye olde Euclidean geometry, I write down an expression for the quantity to be minimized in terms of a chord of central angle $\theta$ (see https://en.wikipedia.org/wiki/Circular_segment for formulae and pictures): $$f(\theta)=\frac{2\sqrt{2-2\cos(\theta)}}{\theta - \sin(\theta)}$$
However, upon plotting this function (http://www.wolframalpha.com/input/?i=%5Cfrac%7B2%5Csqrt%7B2-2%5Ccos(%5Ctheta)%7D%7D%7B%5Ctheta+-+%5Csin(%5Ctheta)%7D), I notice a singularity at zero, and I have no idea how to continue, or where I went wrong…any takers?
This is exactly the behavior you expect. Remember that a central angle of zero would correspond to an infinitely short chord of the circle, while a central angle of $\pi$ gives you a diameter. And indeed you can check that $f(\theta)$ is decreasing on $(0, \pi]$ and hence $$\inf_{E\text{ a chord}} \frac{S(E)}{\min(V(A),V(B))} = f(\pi) = \frac{4}{\pi}$$ as desired.