Positive bases, cones, Helly type theorems

Bárány, Imre (2024) Positive bases, cones, Helly type theorems. MATHEMATICA SLOVACA, Accept. ISSN 0139-9918

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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 $\C^*\subset \C$ of size at most max{d+1,2(d−k+1)} the set $\bigcap \C^*$ contains a k-dimensional cone, then so does $\bigcap \C.$ One ingredient in the proof is another Helly type theorem about the dimension of lineality spaces of convex cones.

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