$\newcommand\conv{\operatorname{conv}}\newcommand\ext{\operatorname{ext}}\newcommand\p{\partial}$The answer is yes. Indeed, let $K:=\conv P$ (the convex hull of $P$), let $\p K$ be the boundary of $K$, and let $\ext K$ be the set of all extreme points of $K$. Then 
$$P\subseteq\conv P=K=\conv\ext K \\ 
=\conv(P\cap\ext K)
\subseteq\conv(P\cap\p K)=\conv C.\quad\Box$$