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