Planar compact sets whose intersections are starshaped via orthogonally convex paths

Abstract

Abstract A Helly-type theorem previously established for a finite family of simply connected orthogonal polygons may be extended to a family of planar compact sets having connected complements: If every three (not necessarily distinct) members of have a nonempty intersection that is starshaped via orthogonally convex paths, then all members of have such an intersection. When is finite, an analogous result holds with orthogonally convex paths replaced by staircase paths. The number three is best possible in each case. Moreover, the results fail without the requirement that the sets have connected complements.