# Prove that the convex hull of a set is the smallest convex set containing that set

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.

• That depends: how do you define the convex hull of $A$? – Gerry Myerson Oct 5 '11 at 2:54
• What is your definition of the convex hull? And don’t you mean the smallest convex set containing A? – Brian M. Scott Oct 5 '11 at 2:54
• This is related. – Did Oct 5 '11 at 5:13

• @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 5 '11 at 4:05