2
$\begingroup$

I'm having trouble understanding the definition of convex hull. Can somebody give me an example? For example if I have the real convex set $(a,b)$ then what is its convex hull? Is it $(a,b)$ or is it $[a,b]$?

Does convex hull of convex set $S$ always equal the convex set $S$ itself? If not, example?

Thank you for any help =)

$\endgroup$
2
$\begingroup$

Convex hull of $A$ is, by definition, the smallest* convex set which contains $A$.

So the convex hull of $(a,b)$, which is already convex by the way, is obviously $(a,b)$.

Similarly, convex hull of any convex set $S$ must be $S$, by definition.

*smallest: Smallest here means that the convex hull is the intersection of all convex sets containing $A$.

$\endgroup$
  • $\begingroup$ aah...now I got it :D so $A$ doesn't necessarily have to be convex set itself :) I thought at first that $A$ has to be convex, but that's just a special case? $\endgroup$ – jjepsuomi Jul 7 '14 at 7:25
  • $\begingroup$ No, $A$ doesn't have to be convex; for example, see the following picture: upload.wikimedia.org/wikipedia/commons/thumb/d/de/… The convex hull of the points (we called this $A$) is the set bounded by the blue line. (Of course there are a lot of handwaving of mathematical rigorousness here, but hopefully you get what I mean) $\endgroup$ – Ivan Wangsa Jul 7 '14 at 7:27
  • $\begingroup$ Got it =) Thank you very much! =) $\endgroup$ – jjepsuomi Jul 7 '14 at 7:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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