Helly-Type Theorems for Countable Intersections of Planar Sets Starshaped via Polygonal k-Paths

Abstract. Let k be a fixed integer, k ≥ 1, and let K be a family of simply connected sets in the plane. 1) If every countable intersection of members of K is starshaped via (at most) k-paths and the corresponding kth-order kernel has nonempty interior, then ∩{K: K inK} has these properties as well. 2) When members of K are closed, if every countable intersection of members of K is starshaped via (at most) k-paths, then ∩{K: K inK} is starshaped via k-paths, also