Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$?

(To avoid technicalities, let's require that the boundary is a 1-dimensional smooth submanifold.)

If that fails, does there exists a set, say, with area between $\pi$ and $2\pi$ and length of the boundary no greater than that of the spherical cap of the same area? Or at least no greater than $2\pi$?

More generally, I am interested in any results saying that the isoperimetric profile of the round metric on the sphere is maximal in some sense (among all Riemannian metrics of the same area).

**Notes.**

The answer is affirmative for central symmetric metrics (i.e. if the metric admits an $\mathbb{RP}^2$ quotient). This follows from Pu's isosystolic inequality: in $\mathbb{RP}^2$ with a metric of area $2\pi$ there exists a non-contractible loop of length at most $\pi$. The lift to the sphere is a loop of length at most $2\pi$ dividing the area in two equal parts.

One should not require that the set is bounded by a single loop. A counter-example is the surface of a small neighborhood of a tripod (formed by three long segments starting from one point) in $\mathbb R^3$. Here one can divide the area in half by two short loops, but one loop would be long. (However one can cut off 1/3 of the area by one short loop.)

In $S^3$ the similar assertion is false, moreover the minimal area of the boundary of a half-volume set can be arbitrary large.