I posted a question on math.stackexchange.com but it seems this question might be open https://math.stackexchange.com/questions/109752/line-segments-intersecting-jordan-curve

Namely,

is there a set $A\subset \mathbb{R}^2$ such that

- The boundary of $A$, $\partial A$, is a Jordan curve and
- For any $B\in \operatorname{int} A\ne\emptyset $, $C\in \operatorname{ext} A\ne\emptyset$ , the line segment $BC$ intersects $\partial A$ infinitely many times?

In the link Leonid Kovalev gave an example that might solve the problem but I have no idea is that an example of such curve. Can anyone verify if the Julia set he gave in the link solves the problem?

anymodulus of continuity, you can construct the jordan curve in such a way that every curve with the given modulus of continuity joininng the inside of the region to the outside intersects the Jordan curve infinitely often. So you can replace "line segment" by Lipschitz continuous curve or Holder continuous curve if you like. $\endgroup$