As I understand it, Caratheodory's Theorem of Convex sets essentially states
If $Q$ is a set in a vector space of dimension $n$ and x lies in the convex hull of $Q$, then x can be written as a convex combination of no more than $n+1$ points in $Q$.
Is this a biconditional statement? I.e. if x can be written as a convex combination of no more than $n+1$ points in $Q$, then is x required to lie in the the convex hull of $Q$?
Any sources and examples/counter-examples would be greatly appreciated!
The convex hull of $Q$ is the smallest convex set of $\mathbb R^n$ that contains every point in $Q$. It is the intersection of all convex sets that contain every point of $Q$. So what is a "convex" set? By one often seen definition, it is a set closed under drawing line segments. A set is convex precisely if, for any two points $A$ and $B$ in the set, every point on the segment from $A$ to $B$ is in the set.
"Convex combination" is often defined as a linear combination in which the sum of the coefficients is $1$ and every coefficient is non-negative.
It's easy to see that the line segment consists of the convex combination of two points. The problem now is: what if there are more than two? Could one get a bigger set by closing under convex combinations of more than two points? I think as long as that number of points is finite, one can prove simply by induction on the number of points that one doesn't get any more.
