Colorful positive bases decomposition and Helly-type results for cones
Grigory Ivanov
TL;DR
The paper develops a colorful extension of Helly-type results for cones by introducing a Colorful Reay decomposition and a Colorful Carathéodory theorem for positive bases. It proves that for k in [d-1], a d+(d-k)+1 coloring of nonzero vectors in R^d yields that if every rainbow sub-selection up to size m(k,d)=max{d+1,2(d-k+1)} has at least k independent solutions to ⟨a,x⟩≤0, then some color class already imposes this bound, with both the Helly number and the number of colors proven optimal. The methods hinge on translating between homogeneous systems and cone intersections via polar duality, and on a constructive inductive decomposition (Colorful Reay decomposition) that organizes the color sets into a nested, minimally positively spanned structure. The results extend to a nonhomogeneous setting, yielding weaker colorful Helly-type statements for cones by reduction to the homogeneous framework. These contributions illuminate intricate connections between positive bases, Reay decompositions, and quantitative Helly-type phenomena in high-dimensional convex geometry.
Abstract
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 Carathéodory-type result for positive bases.