Star-convex is not convex?

Question

Could anyone give a subset which is star-convex in $\mathbb{R^{2}} $ but is not convex?

Answer 1

A five pointed star is a good example.

Every point can be joined to the center by a line segment. However the line joining two adjacent summits is not included in the five pointed star.

Answer 2

A simple example is to take the open unit ball around $0$ and remove the segment $(\frac 1 2 ,1)$ lying on the $x-$ axis.