# How close are star-convex sets to convex sets?

What interesting properties of convex sets are retained by star-convex sets?

-

• The interior of a star-convex set need not be connected. An example is $\mathbb{C} \setminus \lbrace x+iy \;\vert\; x = 0 \text{ and } y \neq 0 \rbrace$, i.e. the right half-plane plus the left half-plane plus the origin.
+1 on the bit about the misnomer. Since convex $\implies$ star, it is very strange to add "star" as a qualifier over "convex". – Willie Wong Dec 12 '10 at 14:09
@Kara, instead of thinking about intervals, you should think about line segments. These can be parametrized by intervals by using vectors, e.g. $[x,y] = [3,1]t+[0,-1]$. So, a star is not convex because the line between the points is not contained inside it. – Eric Stucky Jun 26 '13 at 0:57