Colorful positive bases decomposition and Helly-type results for cones

We prove the following colorful Helly-type result: Fix k ∈ [d-1]. Assume A₁, , Ad+(d-k)+1 are finite sets (colors) of nonzero vectors in ᵈ. If for every rainbow sub-selection R from these sets of size at most max +1, 2(d-k+1)\, the system a,x ≤ 0,\; a ∈ R has at least k linearly independent solutions, then at least one of the systems a,x ≤ 0,\; a ∈ Aᵢ, i ∈ [d+(d-k)+1] has at least k linearly independent solutions. A rainbow sub-selection from several sets refers to choosing at most one element from each set (color). The Helly-number max +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.