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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?

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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 $\mathbb{R}^d$.

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    the disc with one point removed is not a convex domain!2012-04-15
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    The disc with one (interior) point removed is not convex, either, right?2012-04-15
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    You can check using the definition do convex domain!!!2012-04-15
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    Okay I missed the convex assumption;( Yes convex open sets are connected and simply connected.2012-04-15
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    @late_learner can you give me a proof? thank you!2012-04-15
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    "Describe all the convex domains"? What exactly do you mean by that? In any case, this looks rather hopeless to me.2012-04-15
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    Okay t.b., I edited my answer.2012-04-15