can anybody help me please? Is there a good way to prove that given a set of points, say $S = \{x_1, x_2, ..., x_n\}$, then show that the convex hull of $S$, that is, $conv(S)$ contains all the extreme points in $S$?
Is this equivalent to saying that taking the convex hull of the set does not add any extra extreme points?
Thanks a lot.
The easiest way to prove your first question is to show the characterization $$\mathrm{conv}(S) = \bigg\{\sum_{i=1}^n \alpha_i \, x_i : \sum_{i=1}^n \alpha_i = 1, \alpha_i \ge 0\bigg\}.$$ That is, the convex hull just consists of all convex combinations of elements from $S$. Using this characterization, one can easily show that the extreme points of $\mathrm{conv}(S)$ belong to $S$.
To address your second question: as far as I known, extreme points are only defined for convex sets.