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 \( \{K_m : m \geq 1\} \) of \( \mathcal{K} \), assume that \( \bigcap \{K_m : m \geq 1\} \) 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, \( n \geq 1 \), an analogous result holds for sets starshaped via staircase \( n \)-paths.