Positive bases, cones, Helly-type theorems

Abstract

Assume that k ≤ d is a positive integer and 𝓒 is a finite collection of convex bodies in ℝ d . We prove a Helly-type theorem: If for every subfamily 𝓒* ⊂ 𝓒 of size at most max{d + 1, 2(d – k + 1)} the set ⋂ 𝓒* contains a k-dimensional cone, then so does ⋂ 𝓒. One ingredient in the proof is another Helly-type theorem about the dimension of lineality spaces of convex cones.