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: $$h(\gamma) = \frac{l}{\text{vol}(M_\gamma)}.$$ Where $M\setminus \gamma$ denotes the submanifold of minimal volume of $M$ with boundary $\gamma$.

Claim 1 Solutions to the $l$-isoperimetric problem on the closed disk $D$ 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: $$h(M) = \inf_{\gamma \subset M}\left\{\frac{\text{length}(\gamma)}{\text{vol}(M_\gamma)}\right\}.$$

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 or in the literature, 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?

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: $$h(\gamma) = \frac{l}{\text{vol}(M_\gamma)}.$$ Where $M\setminus \gamma$ denotes the submanifold of minimal volume of $M$ with boundary $\gamma$.

Claim 1 Solutions to the $l$-isoperimetric problem on the closed disk $D$ 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: $$h(M) = \inf_{\gamma \subset M}\left\{\frac{\text{length}(\gamma)}{\text{vol}(M_\gamma)}\right\}.$$

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 or in the literature, 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?

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: $$h(M) = \inf_{\gamma \subset M}\left\{\frac{\text{length}(\gamma)}{\text{vol}(M\setminus \gamma)}\right\}.$$

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 or in the literature, 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?

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: $$h(\gamma) = \frac{l}{\text{vol}(M_\gamma)}.$$ Where $M\setminus \gamma$ denotes the submanifold of minimal volume of $M$ with boundary $\gamma$.

Claim 1 Solutions to the $l$-isoperimetric problem on the closed disk $D$ 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: $$h(M) = \inf_{\gamma \subset M}\left\{\frac{\text{length}(\gamma)}{\text{vol}(M_\gamma)}\right\}.$$

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 or in the literature, 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?