Q:

Conditions for the existence of a conical combination of some given vectors such that it lies in a cone?

Let $v_1,v_2,\dots,v_n,u_1,u_2,\dots,u_r\in\mathbb{R}^n$. Can one find analytical conditions (not write the problem up as a convex optomisation problem and argue it can be solved this way) under which there exists $\alpha_1,\alpha_2,\dots,\alpha_r>0$ such that
$\sum_{i=1}^r\alpha_iu_i\in K=\{\sum_{j=1}^n\theta_jv_j:\theta_j>0\}$.
That is, conditions such that there exists some (strictly?) conical combination (sorry, not really sure what the term for this is) of vectors $u_1,\dots,u_r$ that lies in the inside of the cone $K$ defined by $v_1,\dots,v_n$. Clearly, sufficient conditions are that for some $i$, $u_i\in K$, I'm looking for something a bit more insightful (ideally necessary and sufficient).
Thanks.

A:

A first idea: It's about the intersection of two open cones, the first being $K$ and the second $C:=\{\sum_{i=1}^r \alpha_i u_i\ |\ \alpha_i>0\ (1\leq i\leq r)\}$. When they don't intersect (apart from $0$) there is a separating hyperplane, i.e., a nonzero vector $a$ such that $a\cdot u_i\geq 0$ for all $i$ and $a\cdot v_j\leq 0$ for all $j$.

