simple-connectedness of convex and star domains Let $D$ be a convex domain in the complex plane, and is domain $D$ a simply connected domain? What about when $D$ is a star domain?

Yes star-domains are simply connected, since every path is homotopy equivalent to one going through the center.
The disc with one point removed is not simply connected, but also not convex.
Open convex sets are among the star-domains.
All that is not special to the $\mathbb{C}$, but any $\mathbb{R}^d$.
