Consider a closed shape/set $S \in \mathbb{R}^2$ such that there exists an interior point $(x, y) \in S$ (need not be unique) such that any line passing through $(x, y)$ intersects $S$'s boundary at exactly $k$ distinct points.
Is there a name for the family of shapes/sets which follow these constraints for $k=1$ and $k=2$?
I can see that it'll be a superset of convex sets but can't find any specific names.