The Discrete Planar L0-Minkowski Problem

In the discrete setting, the L₀-Minkowski problem extends the question posed and answered by the classical Minkowski's existence theorem for polytopes. In particular, the planar extension, which we address in this paper, concerns the existence of a convex polygonal body which contains the origin, whose boundary sides have preassigned orientations and each triangle formed by the origin with two consecutive vertices is of prescribed area.