Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. Do we always have two points $a, b\in C$ such that the line segment joining $a$ and $b$ lies completely in $\Omega$? What's more, can we have
"For all $\epsilon>0$, there is two points $a ,b\in C$ such that $|a- b| < \epsilon$ and the line segment joining $a$ and $b$ lies completely in $\Omega$".
It seems that it is related to how the $C$ oscillate. If $C$ is some kinds of fractal curves, it might happens that $C$ is nowhere convex. What if we assume that $F$ is of bounded variation?
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