Polynomial valuations on convex functions and their maximal extensions

A unified framework to investigate polynomial valuations on various spaces of convex functions is introduced. It is shown that these different spaces of valuations can be essentially studied as one single class of valuations defined on a particular cone of convex functions. The corresponding extension problem reduces to a single geometric obstruction on the support of these valuations. As an application, explicit integral representations for a subclass of these valuations are established.