TITLE: Caratheodory's theorem for a point in boundary
QUESTION [5 upvotes]: I am wondering whether the following holds: if $x$ in $\mathbb{R}^d$ lies in the boundary of the convex hull of a set $P$, then $x$ can be expressed as a convex combination of $d$ points in $P$.
We know that $d+1$ points in $P$ are enough to express $x$ as their convex combination (Carathéodory's theorem).
If $x$ is in the boundary, aren't $d$ points enough? I have tried many graphical examples but haven't disproved this.

REPLY [6 votes]: Yes, this is correct. Let $C$ be the convex hull of $P$. Since $x\in\partial P$, there is a supporting hyperplane for $C$ passing through $x$; that is, a hyperplane $M$ such that $C$ is disjoint from one of the open half-spaces that $M$ determines. 
Consider a convex combination of elements of $P$ that is equal to $x$. It cannot involve any elements of $P$ that are not in $M$, because then the combination would not be in $M$. Therefore, $x$ is in the convex hull of $P\cap M$. Apply the Carathéodory theorem to $P\cap M$, considered as a subset of $(d-1)$-dimensional space $M$.