Let $C\subseteq \mathbb{R}^d$ be a convex set and let $M$ be the set of extreme points of $C$ (and thus $C$ is the convex hull of $M$). Let $S\subseteq \mathbb{R}^d$ be another convex set. It is known that $C\cap S$ is also convex.
Let $N$ denote the set of extreme points of $C\cap S$. Is it true that $N\subseteq M\cap S$?
PS. This seems to be a basic question regarding the convex set, while after some effort of research, I cannot find a solution. Any help will be appreciated!