As the title, I think, in a closed, bounded, and convex set $C$ in $\mathbb{R}^n$, $C=\{\sum_{i=1}^{k}\lambda_k x_k:x_i\in C_i, \lambda_i\geq 0, \sum_{i=1}^{k}\lambda_k\leq 1\}$, where $C_i$ is a convex, bounded, and closed set for all $i\in\{1,2,3,...,k\}$, if a point $p$ satisfies the following conditions:
$\exists u,l\in C$ such that $l\leq p\leq u$, where $\leq$ is element-wise operator, that means at least one element in $p$, $l$, $r$, satisfying $l_i<p_i<u_i$ for some $i$.
then, $p$ is not an extreme point.
I want to prove that this statement is true. However, I only can show this may be true by drawing a graph. Can someone give me a hint about how to prove this statement? Thank you!
How about this? $p$ is an extreme point.