Colorful positive bases decomposition and Helly-type results for cones

We prove the following colorful Helly-type result: Fix $k \in [d-1]$. Assume $\mathcal{A}_1, \dots, \mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of nonzero vectors in $\R^d$. If for every rainbow sub-selection $R$ from these sets of size at most $\max \{d+1, 2(d-k+1)\}$, the system $\langle {a},{x} \rangle \leq 0,\; a \in R$ has at least $k$ linearly independent solutions, then at least one of the systems $\langle {a},{x} \rangle \leq 0,\; a \in \mathcal{A}_i,$ $i \in [d+(d-k)+1]$ has at least $k$ linearly independent solutions. A \emph{rainbow sub-selection} from several sets refers to choosing at most one element from each set (color). The Helly-number $\max \{d+1, 2(d-k+1)\}$ and the number of colors $d+(d-k)+1$ are optimal. Our key observation is a certain colorful Carath\'eodory-type result for positive bases.