An intersection property for starshaped sets in the plane

Let be a family of compact starshaped sets in the plane. If every three and every two members of have a union which is connected and simply connected, then {F:F in } is simply connected and nonempty. Of course, if every three and every two members of have a starshaped union, the same result holds.Supported in part by NSF grants DMS-8705336, DMS-8908717 and by a Senior Faculty Summer Research Fellowship, Research Council, University of Oklahoma.