On the existence of U-polygons of class c≥4 in planar point sets

AbstractFor a finite set U of directions in the Euclidean plane, a convex non-degenerate polygon P is called a U-polygon if every line parallel to a direction of U that meets a vertex of P also meets another vertex of P. We characterize the numbers of edges of U-polygons of class c≥4 with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets