# Understanding convex hull

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 =)

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 $A$.
• 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? Jul 7, 2014 at 7:25
• 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) Jul 7, 2014 at 7:27