Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 5 down vote accepted

"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.
share|improve this answer
1  
+1 on the bit about the misnomer. Since convex $\implies$ star, it is very strange to add "star" as a qualifier over "convex". –  Willie Wong Dec 12 '10 at 14:09
    
Thanks @kahen. Excellent answer to a vague and underdetermined question. –  isomorphismes Dec 22 '10 at 7:07
    
Also, how does convex work in R^2? Let Y be a subset of R^2 (the plane). If you have a point x and y with x < y, then the interval (x, y) must also lie in Y (for Y to be convex in X). What is the interval (x, y) if x and y are points on the plane? –  Kara May 17 '13 at 17:11
    
Why is a "star" in the plane not convex? I see that it is star convex, but why isn't it convex? –  Kara May 17 '13 at 17:33
    
@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. –  Eric Stucky Jun 26 '13 at 0:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.