Extreme Points of a Convex Hull Given an arbitrary set $A\subset\mathbb R^n$, why do all extreme points of the convex hull $\operatorname{conv}(A)$ lie in $A$?
(An extreme point of a convex set is defined as one that cannot be written as a strictly convex combination of two distinct points of this set.)

Let $x\in \text{conv}(A)$ be extreme. By definition of the convex hull, $x=\sum_i \lambda_i a_i$ for $a_i\in A$, $\sum_i \lambda_i=1$, and $\lambda_i\ge 0$. Since $A\subset\text{conv}(A)$ and $x$ was extreme, only one $\lambda_i$ can be nonzero (and it must be equal to $1$), so $x=a_i$ for this $i$. Thus $x\in A$.
