Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.

Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?

It is easy to find an open disc which boundary lies in a plane, but the boundary might be crazy; for example it might be Polish circle shown on the diagram.

in the plane?" $\endgroup$ – Joseph O'Rourke Dec 16 '16 at 20:02