I am studying the book "matching theory" by Lovasz and Plummer, and I found the following statement (page 257):
Comparing it with Caratheodory's theorem in Wikipedia reveals two differences:
- The book speaks about vectors in a cone, particularly, in the conic hull of some given vectors. Wikipedia speaks about vectors in the convex hull of some given vectors.
- The book says that $n$ vectors are sufficient, where $n$ is the dimension of the space. Wikipedia says that $n+1$ vectors are sufficient.
I could not find an explicit statement of Caratheodory's theorem for the conic hull elsewhere. Does it easily follow from the theorem for convex hull?
I presume that if states that if a cone $C$ in $\Bbb R^n$ (or any $n$-dimensional real vector space) is generated by a set $S$, then each element of $C$ is a non-negative linear combination of at most $n$ vectors in $S$.
I'm not sure anyone else uses the term conic hull but it makes sense to define it as the set of non-negative linear combinations of the vectors in question.
One can prove this Caratheodory-type result by the same method as the usual one. Let $u\in C$ and suppose $u=\sum_{i=1}^m a_i v_i$ is a minimal representation of $u$ with $a_i>0$ and $v_i\in S$. If $m>n$ then there is a linear dependence $0=\sum_{i=1}^m b_i v_i$ and then for a suitable choice of $\lambda$, $u=\sum_{i=1}^m (a_i+\lambda b_i) v_i$ has non-negative coefficients one of which vanishes, contradicting minimality.