This theorem is obviously true if the set $X$ is finite (so that $\mathop{\rm conv} X$ is a convex polytope). I believe it is true for any set $X\subseteq\mathbb{R}^n$ but I cannot prove it. Can anybody please prove this or give a counter-example? Many thanks!

Notes: $\mathop{\rm conv}X$ denotes the convex hull of the set $X$. A face of a convex set $C$ is defined to be a convex set $F\subseteq C$ such that every line segment from $C$ whose relative interior has a non-empty intersection with $F$ is contained in $F$.