A solution to the **$l$-isoperimetric problem** on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant:
$$h(\gamma) = \frac{l}{\text{vol}(M\setminus \gamma)}.$$
With abuse of notation, $M\setminus \gamma$ denotes the submanifold of minimal volume of $M$ with boundary $\gamma$.

**Claim 1** Solutions to the $l$-isoperimetric problem on the closed disk equipped with a radially symmetric metric are radially symmetric.

My disappointment for not finding a proof for such an obvious statement has been dwarfed only by my surprise in finding a counterexample: Claim 1 is false. (The idea of the counterexample is to have a _small_ flat metric in a ball centered in the origin and a _large_ flat metric in a disjoint annulus, then take $l$ small enough.)

My counterexample seems to fundamentally rely on two factors: the metric can be increasing, and the length $l$ can be chosen to be small. Taking care of both problems at the same time leads to the next claim.

A solution to the **Cheeger-isoperimetric problem** on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ which realizes the [Cheeger constant][1]:
$$h(M) = \inf_{\gamma \subset M}\left\{\frac{\text{length}(\gamma)}{\text{vol}(M\setminus \gamma)}\right\}.$$

**Claim 2** Solutions to the Cheeger-isoperimetric problem on the closed disk equipped with a monotone decreasing radially symmetric metric are radially symmetric.

Statements similar to Claim 2 have been discussed on [Math.OF][2] or in the [literature][3], but the proposed claim doesn't seem to be covered by these sources. **Is Claim 2 correct?** Is it obvious? If correct, does it remain so dropping the monotonicity hypothesis? 


  [1]: https://en.wikipedia.org/wiki/Cheeger_constant
  [2]: http://mathoverflow.net/questions/100666/for-what-metrics-are-circles-solutions-of-the-isoperimetric-problem
  [3]: http://arxiv.org/pdf/1002.1829.pdf