Where does this proof of Caratheodory's theorem break down for using n vectors, instead of n+1?

Question

In the referenced text below, (1) is referring to Caratheodory's theorem for conical combinations.

For Caratheodory's theorem for convex combinations, I'm trying to understand what the purpose is of the up-projection. Now intuitively and from counterexamples I know that just n vectors doesn't suffice for convex combinations. However, it seems to me that if we just don't do the up-projection and go through the proof, we can still invoke (1) and go through the proof as normal...? (I'm sure I am mistaken, but not sure why.)

Any thoughts appreciated.

Thanks.

Answer 1

The reason is that if you take $X$ to be the vertices of a full-dimensional simplex in $\mathbb{R}^n$, then $|X|=n+1$ and the vector which is their average cannot be represented as a convex combination of less than $n+1$ vectors taken from $X$.