What star domain has a non-star-domain interior? 
Definition: We call a subset $S$ of $\mathbb{R}^n$ a star domain (or star-shaped) if there exists a point $x_0 \in S$ such that for every $x \in S$, the line segment $\overline{x_0x}$ is contained in $S$.

The Wikipedia page on star domains claims that

The closure of a star domain is a star domain, but the interior of a star domain is not necessarily a star domain.

I must be particularly unimaginative, as I find myself unable to come up with an example of a star domain whose interior is not a star domain. What is such an example?
 A: For example, $\{(x,y)\in\mathbb R^2\mid xy\ge 0\}$ (that is, the first and third quadrant of the plane, plus the coordinate axes). This is a star-domain with star center $(0,0)$, but its interior is not even connected, and so cannot be a star-domain.

Whenever the original star-domain has an interior point as star-center, its interior will still be a star-domain: Assume without loss of generality that $x_0=0$, and suppose that $x$ is an interior point of $S$. Then it has a neighborhood entirely in $S$. For every $t\in (0,1]$ we can scale this entire neighborhood by a factor of $t$ get a neighborhood of $tx$ which must be entirely in $S$ due to the star-domain property. So we know that every point on $\overline{x_0x}$ must be in the interior of $S$, except possibly $x_0$ itself. Thus, if only $x_0$ is an interior point, then it will work as a star-center for the interior of $S$ too.
On the other hand, this is not a necessary condition: There are some star-domains whose only star-center is a boundary point, but whose interior is nevertheless a star-domain -- for example, the union of the two coordinate axes and one of the quadrants of the plane.
