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How do you prove that the convex hull of A is the smallest convex set containing A?

edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called the convex hull of A.

Rodrigo de Azevedo
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Jeremy Yeo
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1 Answers1

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Young, when you wrote

How do you prove that the convex hull of A is the smallest set containing A?

You meant that convex hull of A is the minimal convex set containing A, right?

To show this, which part is your definition? The linear-algebraic characterization?

You can see that any intersection of convex sets containing A is also a convex set containing A.

Nathan Portland
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  • Yea, the convex hull of A is the minimal convex set containing A. I guess what I need to prove is that the convex hull of A is itself a convex set containing A. How do I prove that or is it just by definition of a convex hull? – Jeremy Yeo Oct 05 '11 at 03:55
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    @Young: Let $\mathscr{C}$ be the collection of all convex sets containing $A$, and let $K=\bigcap\limits_{C\in\mathscr{C}}C$; certainly $K\supseteq A$, and you need to show that $K$ is convex. Let $x,y\in K$ and $t\in[0,1]$; you need to show that $tx+(1-t)y\in K$. Use the facts that (1) $x,y\in C$ for every $C\in\mathscr{C}$ and (2) every $C\in\mathscr{C}$ is convex. – Brian M. Scott Oct 05 '11 at 04:05