Carathéodory's theorem to show point must be in convex hull

Question

I want to use this theorem to show that some point must be contained in the convex hull. But I'm not sure if the theorem says that if I can express the point as a convex combination of the contained set then that point is in the convex hull. The theorem is below

Thanks in advance

Each point in the convex hull of a set $S$ in $\mathbb{R^n}$ is in the convex combination of $n+1$ or fewer points of $S$.

Answer 1

Not sure whether I understand your question correctly.

If a point $p$ is a convex combination of points $x_k\in S$, i.e., $$p=\sum_{k=1}^N \lambda_kx_k\quad {\rm with}\quad \lambda_k\geq0 \ (1\leq k\leq N), \quad\sum_{k=1}^N\lambda_k=1\ ,$$ then $p$ is in the convex hull of $S$. This is an immediate consequence of the definition of convex hull – you don't need Caratheodory's theorem for that.