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

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"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:

  • The intersection of two star-convex sets need not be connected (and thus in particular not star-convex). You can take two L-shaped sets and their intersection will be two disjoint squares.
  • The interior of a star-convex set need not be connected. An example is $\mathbb{C} \setminus \lbrace x+iy \;\vert\; x = 0 \text{ and } y \neq 0 \rbrace$, i.e. the right half-plane plus the left half-plane plus the origin.
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    @Kara, instead of thinking about intervals, you should think about line segments. These can be parametrized by intervals by using vectors, e.g. $[x,y] = [3,1]t+[0,-1]$. So, a star is not convex because the line between the points is not contained inside it.2013-06-26