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?