Helly-type theorems in $R^d$ for infinite intersections of sets starshaped via staircase paths

Abstract

Let $\mathcal{K}$ be a family of sets in ${\mathbb{R}}^d$. For each countable subfamily $\{K_m:m\ge 1\}$ of $\mathcal{K}$, assume that $\bigcap \{K_m:m\ge 1\}$ is consistent relative to staircase paths and starshaped via staircase paths, with a staircase kernel that contains a convex set of dimension $d$. Then $\bigcap \{K:K\in \mathcal{K}\}$ has these properties as well. For $n$ fixed, $n\ge 1$, an analogous result holds for sets starshaped via staircase $n$-paths.