How one can prove that convex hull is the minimal convex set containing $X$?
We need to show that for each convex set $M$ if $X\subseteq M$ then $conv(X)\subseteq M$.
I am thinking of proof by contradiction. Let $x\in conv(X)$ but $x \notin M$, then we can separate $x$ from $M$. How to get the contradiction?
Lets define convex hull in this way:
$conv(X) = \{x\ |\ x=\alpha_1x_1+\dots\alpha_kx_k,\ \alpha_i\ge0(1\le i\le k),\ x_i\in X,\ \alpha_1+\dots\alpha_k=1,\ k\in\mathbb N \}$