Have a very basic question on closed plane curves, from this Wikipedia math reference desk conversation, that I can't seem to come up with any results for. In particular:
If $C : [0, 1] \rightarrow \mathbb{R}^{2}$ is a closed curve (not necessarily simple), then does there necessarily exist a line $L$ such that $L$ intersects the image of $C$ at exactly one point?
If the question is rephrased to two intersections, then since simplicity is not required, space-filling curves serve as effective counterexamples. And, if the question is rephrased to two intersections plus simplicity, then it appears that the boundary of the dragon curve (or some dragon-like curve) serves as a counterexample.
The function $C(I)\times C(I)\to \mathbb R_{\ge0},\;\; (x,y)\to d(x,y)=|x-y|$ is continuous, moreover the domain is compact because its product of compact space ($C(I)$ is the image of compact set $[0,1]$ via the continuous map $C$). Then exists $\bar x,\bar y$ s.t. $d(\bar x,\bar y)$ is maximum, hence $C(I)$ is inside the circumference of centre $\bar x$ and ray $d(\bar x,\bar y)$.
Now we are done because we have just to take the line tangent to that circumference in $\bar y$.