Is the intersection of two star-shaped sets star-shaped?
I don't think so but can't think of an example.
Is the intersection of two star-shaped sets star-shaped?
I don't think so but can't think of an example.
Yes, it can be.
No, it does not have to be.
HINT: A star-shaped set is connected. A $+$ sign is star-shaped. Find a way to position two of them so that the intersection is a two-point set.
Intersection of two star shaped domain is again a star shaped domain provided previous two domains are star shaped with respect to same point. Otherwise the result is not true in general.See the counterexample given by Scott.