Positive bases, cones, Helly type theorems

TL;DR

This paper establishes a Helly type theorem for convex bodies in Euclidean space, linking the intersection properties of subfamilies to the existence of a common cone of a specified dimension.

Contribution

It introduces a new Helly type theorem connecting convex intersections with the presence of a common convex cone of a given dimension.

Findings

Proves a Helly type theorem for convex bodies and cones in d.

Provides a Helly theorem related to the dimension of lineality spaces.

Establishes bounds on subfamily sizes for the theorem to hold.

Abstract

Assume that $k \le d$ 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.