Convex decomposition of U-polygons

When parallel X-rays are considered in any finite set U of directions, switching components with respect to U can be constructed. This is true for U⊂Rn, and also for any finite set of lattice directions. R.J. Gardner raised the problem of looking for a characterization of switching components. In 2001, L. Hajdu and R. Tijdeman gave an answer by proving that a switching component is always the linear combination of switching elements. Though splendid, this result fails to be a characterization theorem inside the class of convex bodies, meaning that the switching element of the linear combination could be not convex even if the switching component is convex. The purpose of this paper is to investigate the problem in the plane, where a convex switching component with respect to U is a U-polygon. We prove that a U-polygon can always be decomposed as a linear sum inside the class of U-polygons.