A Helly-type theorem for countable intersections of starshaped sets

Abstract.

Let k and d be fixed integers, 0≦k≦d, and let $\mathcal{K} = \{ k_\alpha :\alpha {\text{ in some index set}}\}$ be a collection of sets in $\mathbb{R}^d .$ If every countable subfamily of $\mathcal{K}$ has a starshaped intersection, then $\cap \{ k_\alpha :k_\alpha {\text{ in }}\mathcal{K}\}$ is (nonempty and) starshaped as well. Moreover, if every countable subfamily of $\mathcal{K}$ has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of $\cap \{ k_\alpha :k_\alpha {\text{ in }}\mathcal{K}\}$ is at least k-dimensional, too. Finally, dual statements hold for unions of sets.