$A \subset [0,1]^n$ is a compact, convex subset of $\mathbb{R}^n.$ Take any subset (Not necessarily connected) $B \subseteq \partial A$, consisting of finite number of disjoint, closed subsets of the boundary of $A$. I want to show that every point in the convex hull of $B$ can be represented as the convex combination of at most two points of $B$.
This seems useful but apparently (according to comments) both the statement and the proof provided are inaccurate. This is the approach I have been taking so far. The part about induction on $n$ didn't occur to me but I tried to proceed using the supporting hyperplane theorem.
Any help is appreciated. Thanks in advance.