The Dimension of the Kernel for Intersections of Certain Starshaped Sets in ${\mathbb{R}}^3$

Abstract

Let $\mathcal{S}$ be a finite family of sets in ${\mathbb{R}}^d$, each a finite union of polyhedral sets at the origin and each having the origin as an extreme point. Fix $d$ and $k$, $0\le k\le d\le 3$. If every $d+1$ (not necessarily distinct) members of $\mathcal{S}$ intersect in a star-shaped set whose kernel is at least $k$-dimensional, then $\cap \{S_i:S_i\in \mathcal{S}\}$ also is a star-shaped set whose kernel is at least $k$-dimensional. For $k\ne 0$, the number $d+1$ is best possible.