Fact: Start with $m$ points $u_1,...,u_m\in \mathbb R^n$. Now pick any $u$ in the cone of $u_1,...,u_m$; that is,
$$ u = \sum_{i=1}^m \alpha_i u_i \text{ for some } \alpha_i \geq 0, \; i = 1,...,m. $$
Then it is known that there exists nonnegative coefficients $\beta_1,...,\beta_m$ such that $$ u = \sum_{i=1}^m \beta_i u_i. $$ where at most $n$ coefficients $\beta_1,...,\beta_m$ are nonzero.
This fact is used to prove many things, such as Caratheodory's theorem and the Shapley-Folkman lemma. It is sometimes called the "conic Caratheodory's theorem."
I've seen this result in two places, one which says the "proof is in many texts on linear programming" and another which uses Caratheodory's theorem to prove it, which I find circular.
Can someone give me a proof of this fact that doesn't use Caratheodory?
To start, we know that if we drop the nonnegativity constraints on the coefficients, then this is a simple property in linear algebra. Specifically, if $u \in \mathbf{span}(u_1,...,u_m)$ and $m > n$, then $u$ can be written as a linear combination of at most $n$ points in $\{u_1,...,u_m\}$. But it's not obvious to me how to extend this to the ``conic version".