How close are star-convex sets to convex sets?

Question

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

Answer 1

"Star-convex set" is a bit of a misnomer. The prevailing term probably should have been "star domain" instead (but star-convex is so common that the ship has sailed on that one) since that is what you use it for: An open star domain is a simply connected domain - a handy fact for proving simple cases of theorems in e.g. complex analysis.

In short, they have very few of the properties of convex sets. For example:

• The intersection of two star-convex sets need not be connected (and thus in particular not star-convex). You can take two L-shaped sets and their intersection will be two disjoint squares.
• 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.