I read wikipedia's proof of Caratheodry's theorem for convex sets. I was wondering if there exists a more geometric proof.
I was thinking of something along the following line of reasoning.
The theorem:
Given a set $P$ of points in $R^d$. A point $x$ in the convex hull can be written as a convex combination of at most $d+1$ points in $P$.
There is a line from a point $p'\in P$ to $x$. This line must intersect the boundary of the polyhedron, say at $y$. (Which is bounded because it is the convex hull of a finite set of points.)
Then $x=\lambda p' + (1-\lambda)y$ where $0 \le \lambda \le 1$
This boundary is a face of the polyhedron which lives on a $d-1$ dimensional subspace. Therefore by induction, $y$ is a convex combination of d points in $p$ and therefore $x$ is a convex combination of $d + 1$ points.
The base case for the induction argument just is that for points in any 1 dimensional subspace, one can just take the two outer points to make a convex combination of any points in the convex set.