# An asymptotic version of the Isoperimetric inequality

Let $$U$$ be a simply connected bounded open set in $$\mathbb{R}^2$$. The area of $$U$$ is denoted by $$A$$.

(We do not assume any thing about its boundary).

Assume that $$\gamma_n$$,s are smooth simple closed curves which lie in $$U$$. The perimiter and area of $$\gamma_n$$ are denoted by $$l_n$$ and $$A_n$$, respectively. We assume that $$\gamma_n$$,s eventually leave compact subsets of $$U$$. That is for every compact subset $$K\subset U$$, there is a natural number $$N$$ such that $$\gamma_n$$ has empty intersection with $$K$$, for every $$n>N$$. Assume that $$A_n$$ converges to $$A$$ and $$l_n$$ converges to a real number $$l$$ and we have$$4\pi A=l^2$$.

Is $$U$$ necessarily the interior of a circle?

I may have missed something but it should follow from Bonnesen's inequality, which states that every domain $$\Omega\subset\mathbb{R}^2$$ satisfies : $$\mathcal{L}(\partial\Omega)^2-4\pi\mathcal{A}(\Omega)\geq \pi^2(r_\text{ex}(\Omega)-r_\text{in}(\Omega))^2$$ where $$r_\text{in}(\Omega)$$ (resp. $$r_\text{ex}(\Omega)$$) is the biggest( resp. smallest) possible radius of disk contained in $$\Omega$$ (resp. which contains $$\bar\Omega$$).
If one denotes by $$\Omega_n$$ the domain bounded by your $$\gamma_n$$. Then your hypothesis imply that each of the $$\Omega_n$$ are sandwiched between two disks of closer and closer radius, which should be enough to conclude.