Under the axiom of choice, there exists a set $A\subseteq \mathbb{R}^2$ which intersects every line in exactly two points (See here). As I understand, it is not clear whether the axiom of choice is necessary.
Let us consider the following weaker statement: There exists a set $A\subseteq \mathbb{R}^2$ which satisfy both of the following conditions:
- $A$ intersects every line in at most two points, and
- For every $(x,y)\in \mathbb{R}^2$ there are pair of points in $A$, such that $(x,y)$ sits on the line which connects them.
This surely holds under the axiom of choice. Is it known whether this weaker version holds witout choice?
Originally from the comments:
The answer is yes: consider any nontrivial circle in the plane. More generally, any smooth strictly convex closed curve does this.
It is not immediately clear how to strengthen the conditions to avoid this without getting an immediate negative answer.