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 ${\mathbb{R}}^d$. If for every rainbow sub-selection $R$ from these sets of size at most $\max \{d+1,2(d-k+1)\}$, the system $⟨a,x⟩\le 0,a\in R$ has at least $k$ linearly independent solutions, then at least one of the systems $⟨a,x⟩\le 0,a\in {\mathcal{A}}_i,$ $i\in [d+(d-k)+1]$ 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 Carathéodory-type result for positive bases.