Affiliation:
1. Alfréd Rényi Institute of Mathematics 13-15 Reáltanoda Street Budapest Hungary
2. Department of Mathematics University College London Gower Street WC1E 6BT London United Kingdom
Abstract
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.
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