Polynomial valuations on convex functions and their maximal extensions

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the support of these valuations. The results rely on a homogeneous decomposition for the space of polynomial valuations of bounded degree and the support properties of certain distributions associated to the homogeneous components. As an application, an explicit integral representation for valuations of top degree is established.