On the geometric structure of lattice U-polygons

Let U be a finite set of directions in the Euclidean plane. A convex polygon P is a U-polygon if for each vertex v of P   and for each u∈Uu∈U the line with direction u through v meets a vertex of P different from v. We study the geometric structure of lattice U-polygons and introduce the notion of class of a U-polygon. We then characterize the lattice U  -polygons of class c⩾4c⩾4. On the other hand, if P is a lattice U  -polygon of class c<4c<4, we describe a few geometric properties of P.